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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

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Deflationism and Axiomatic Theories of Truth

Proof theoretic and model theoretic conservativities Stella Moon

ILLC, UvA

2nd October 2015

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Deflationism

Truth is insubstantial so it does not carry any ontological weight.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Deflationism

Truth is insubstantial so it does not carry any ontological weight. Deflationists are interested how truth works, rather than what it is.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Deflationism

Truth is insubstantial so it does not carry any ontological weight. Deflationists are interested how truth works, rather than what it is. It is true that snow is white iff snow is white.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Deflationism

Truth is insubstantial so it does not carry any ontological weight. Deflationists are interested how truth works, rather than what it is. It is true that snow is white iff snow is white. Motivated by Tarski’s biconditionals: for any sentence φ T(φ) ↔ φ.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Deflationists desire our extended theory to be conservative

ver our base theory.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Deflationists desire our extended theory to be conservative

ver our base theory.

There are two notions of conservativities: for models and for theories.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Definition (Proof theoretic conservativity) Let Γ be a L-theory and Γ′ be a L′-theory extending Γ, that is L′ ⊇ L, such that Γ′ ⊇ Γ. Γ′ is proof theoretically conservative

ver Γ if for any L-setenece θ, Γ′ ⊢ θ, we have that Γ ⊢ θ.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Definition (Proof theoretic conservativity) Let Γ be a L-theory and Γ′ be a L′-theory extending Γ, that is L′ ⊇ L, such that Γ′ ⊇ Γ. Γ′ is proof theoretically conservative

ver Γ if for any L-setenece θ, Γ′ ⊢ θ, we have that Γ ⊢ θ.

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′.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Peano Arithmetic

Axioms of arithmetic (natural numbers)

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Peano Arithmetic

Axioms of arithmetic (natural numbers) – self-reference.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Peano Arithmetic

Axioms of arithmetic (natural numbers) – self-reference. La = {<, +, ·, s, 0}

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Peano Arithmetic

Axioms of arithmetic (natural numbers) – self-reference. La = {<, +, ·, s, 0} G¨

del’s diagonal Lemma and the incompleteness theorems.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Axioms of Peano Arithmetic (PA)

∀x(s(x) = 0) ∀x, y(s(x) = s(y) → x = y) ∀x(x + 0 = x) ∀x, y(x + s(y)) = s(x + y) ∀x(x · 0 = 0) ∀x, y(x · s(y) = (x · y) + x) ∀x(¬x < 0) ∀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))
→ ∀xφ(x)
.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Models of arithmetic

Standard model: N = N; <N ; +N , ·N ;sN ; 0N .

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Models of arithmetic

Standard model: N = N; <N ; +N , ·N ;sN ; 0N . Non-standard models: M = M; <M; +M, ·M;sM;0M.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Models of arithmetic

Standard model: N = N; <N ; +N , ·N ;sN ; 0N . Non-standard models: M = M; <M; +M, ·M;sM;0M. There is a c ∈ M such that for any n ∈ N, n < c.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Models of arithmetic

Standard model: N = N; <N ; +N , ·N ;sN ; 0N . Non-standard models: M = M; <M; +M, ·M;sM;0M. There is a c ∈ M such that for any n ∈ N, n < c. They are not isomorphic to each other.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

G¨

del coding

f : FormLa − → N

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

G¨

del coding

f : FormLa − → N f is a recursive function

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

G¨

del coding

f : FormLa − → N f is a recursive function im(f ) ⊆ N is recursive

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

G¨

del coding

f : FormLa − → N f is a recursive function im(f ) ⊆ N is recursive f −1 is a recursive function

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

G¨

del coding

— La-symbols Natural numbers s 1 + 2 · 3 < 4 = 5 ∧ 6 ∨ 7 ¬ 8 ∃ 9 ∀ 10 vi (11, i)

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Theorems

Theorem (Diagonal Lemma) For any formula φ(x), there is a sentence θ such that PA ⊢ φ(θ) ↔ θ.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Theorems

Theorem (Diagonal Lemma) For any formula φ(x), there is a sentence θ such that PA ⊢ φ(θ) ↔ θ. Proof We define a function diag : N → N in the following way: diag(n) = ∀y(y = n → σ(y)), if n = σ(x) for some formula 0,

therwise.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Definition (Provability predicate) For any La formula φ, the provability predicate in PA is defined in the following way. If PA ⊢ φ then PA ⊢ Prv(φ) PA ⊢ Prv(φ → ψ) → (Prv(φ) → Prv(ψ)) PA ⊢ Prv(φ) → Prv(Prv(φ))

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Theorems

Theorem (G¨

del’s first incompleteness theorem)

There is a sentence G such that PA ⊢ G ↔ ¬Prv(G).

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Theorems

Theorem (G¨

del’s first incompleteness theorem)

There is a sentence G such that PA ⊢ G ↔ ¬Prv(G). Theorem (G¨

del’s second incompleteness theorem)

Assume PA is consistent and let ConPA := ¬Prv(⊥) be the sentence defining the consistency of PA. Then PA ConPA i.e. PA cannot prove its own consistency.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Liar sentence

Theorem Assume the Tarski-biconditionals for all sentences in PA. Let T be a predicate defining truth in PA. T is undefinable in La.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Liar sentence

Theorem Assume the Tarski-biconditionals for all sentences in PA. Let T be a predicate defining truth in PA. T is undefinable in La. Proof Assume for a contradiction that T is a definable in La. Then by the Diagonal Lemma, there is a sentence θ such that PA ⊢ θ ↔ ¬T(θ). Then by soundness, N θ ↔ ¬T (θ). But by the TB, N θ ↔ T (θ).

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Typed theories of truth

We cannot assert truth over a sentence containing truth.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Typed theories of truth

We cannot assert truth over a sentence containing truth. It is true1 that it is true0 that snow is white.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Typed theories of truth

We cannot assert truth over a sentence containing truth. It is true1 that it is true0 that snow is white. We don’t have a problem with the Liar sentence anymore.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Typed theories of truth

We cannot assert truth over a sentence containing truth. It is true1 that it is true0 that snow is white. We don’t have a problem with the Liar sentence anymore. Assert truth over sentences in PA.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

The compositional theory of truth

∀s∀t(T(s = t) ↔ val(s) = val(t)) ∀x(sent(x) → (T(¬x) ↔ ¬T(x))) ∀x∀y(sent(x ∧ y) → (T(x ∧ y) ↔ T(x) ∧ T(y))) ∀x∀y(sent(x ∨ y) → (T(x ∨ y) ↔ T(x) ∨ T(y))) ∀v∀x(sent(∀vx) → (T(∀vx) ↔ ∀tT(x(t/v)))) ∀v∀x(sent(∃vx) → (T(∃vx) ↔ ∃tT(x(t/v))))

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Tarski biconditionals are valid in CT

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Tarski biconditionals are valid in CT CT is neither proof theoretically nor model theoretically

conservative. It can prove the consistency of PA.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Tarski biconditionals are valid in CT CT is neither proof theoretically nor model theoretically

conservative. It can prove the consistency of PA.

Solution: Restrict the induction axiom schema from PA, to get CT−.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Tarski biconditionals are valid in CT CT is neither proof theoretically nor model theoretically

conservative. It can prove the consistency of PA.

Solution: Restrict the induction axiom schema from PA, to get CT−. CT− is proof theoretically conservative. (Enayat & Visser (2013) and Leigh (2013))

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Tarski biconditionals are valid in CT CT is neither proof theoretically nor model theoretically

conservative. It can prove the consistency of PA.

Solution: Restrict the induction axiom schema from PA, to get CT−. CT− is proof theoretically conservative. (Enayat & Visser (2013) and Leigh (2013)) Tarski biconditionals are valid in CT−

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Satisfaction classes

Definition (Satisfaction class) S ⊆ N2 is a satisfaction class of a model M if S = {(φ(x), c)|M φ(c)} SM(φ, c) ⇔ M T(φ(c))

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Satisfaction classes

Definition (Satisfaction class) S ⊆ N2 is a satisfaction class of a model M if S = {(φ(x), c)|M φ(c)} SM(φ, c) ⇔ M T(φ(c)) We expand a model M PA by adding the satisfaction class to M to get a model of CT−.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Is CT− model theoretically conservative?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Is CT− model theoretically conservative? No (By Lachlan’s theorem)

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Is CT− model theoretically conservative? No (By Lachlan’s theorem) There are non-standard models of CT− extending PA

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Is CT− model theoretically conservative? No (By Lachlan’s theorem) There are non-standard models of CT− extending PA such that M T((0 = 1) ∨ ... ∨ (0 = 1)). (Kotlarski, Krajewski, Lachlans (1981))

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Proof theoretic and model theoretic conservativities

Is CT− model theoretically conservative? No (By Lachlan’s theorem) There are non-standard models of CT− extending PA such that M T((0 = 1) ∨ ... ∨ (0 = 1)). (Kotlarski, Krajewski, Lachlans (1981)) Call the satisfaction class S that contains arithmetically false sentences to be pathological.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Eliminate pathological satisfaction classes, containing arithmetically false sentences.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Eliminate pathological satisfaction classes, containing arithmetically false sentences. Cieslinski (2011) adds sentences such as ProvPA(x) → T(x).

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Eliminate pathological satisfaction classes, containing arithmetically false sentences. Cieslinski (2011) adds sentences such as ProvPA(x) → T(x). All these theories are not proof theoretically conservative.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Eliminate pathological satisfaction classes, containing arithmetically false sentences. Cieslinski (2011) adds sentences such as ProvPA(x) → T(x). All these theories are not proof theoretically conservative. What can we do?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Tarski’s model theoretic definition of truth?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Tarski’s model theoretic definition of truth? Non-standard models give a non-standard interpretation?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Tarski’s model theoretic definition of truth? Non-standard models give a non-standard interpretation? Second order arithmetic with full-semantics?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Solutions?

Tarski’s model theoretic definition of truth? Non-standard models give a non-standard interpretation? Second order arithmetic with full-semantics? Is model theoretic conservativity for deflationists?

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference. PA cannot define truth.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference. PA cannot define truth. We add the compositional axioms and T predicate to attain CT.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference. PA cannot define truth. We add the compositional axioms and T predicate to attain CT. CT fails to be proof/model theoretically conservative. So we restrict it to CT−.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference. PA cannot define truth. We add the compositional axioms and T predicate to attain CT. CT fails to be proof/model theoretically conservative. So we restrict it to CT−. CT− is not model theoretically conservative, and it states arithmetically false sentences are true.

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Deflationism and Axiomatic Theories of Truth Stella Moon Deflationism Peano Arithmetic The Compositional Theory of Truth

Summary and conclusion

Deflationists desire proof theoretic and model theoretic conservativities. We start with PA as our base theory as it allows self-reference. PA cannot define truth. We add the compositional axioms and T predicate to attain CT. CT fails to be proof/model theoretically conservative. So we restrict it to CT−. CT− is not model theoretically conservative, and it states arithmetically false sentences are true. Cieslinski’s elimination methods fails to save CT−.