[PPT] - Does relative interpretation preserve meaning? Mirko Engler Dept. PowerPoint Presentation

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Does relative interpretation preserve meaning?

Mirko Engler

Dept. of Philosophy, HU Berlin & Dept. of Mathematics, U Nova Lisboa

PhD’s in Logic X

Prague, May 2, 2018

This work is supported by the Portuguese Science Foundation, FCT, through the project PTDC/MHC-FIL/2583/2014. Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 1 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic a lot of important properties of a theory are preserved

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic a lot of important properties of a theory are preserved what can we say about preserving the meaning of a theory?

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic a lot of important properties of a theory are preserved what can we say about preserving the meaning of a theory? we state two thesis of preservation of meaning

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic a lot of important properties of a theory are preserved what can we say about preserving the meaning of a theory? we state two thesis of preservation of meaning we consider the interpretability of inconsistency

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Introduction

relative interpretations had become a useful tool in comparing theories in mathematical logic a lot of important properties of a theory are preserved what can we say about preserving the meaning of a theory? we state two thesis of preservation of meaning we consider the interpretability of inconsistency meaning of theory is not in generel preserved by rel. interpretations

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 2 / 17

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Overview

1

Relative Interpretation Definitions Examples Basic Principles

2

Preservation of Meaning Meaning and Interpretation Preservation of Meaning

3

A Counterexample Expressing Inconsistency Feferman’s Theorem A Counterexample

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 3 / 17

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Definitions

Let L[S] and L[T] be languages of 1st order theories S and T with identity.

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 4 / 17

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Definitions

Let L[S] and L[T] be languages of 1st order theories S and T with identity.

Definition (Relative Translation)

A relative translation f for L[S] to L[T] is given by a pair I, δ, where I : L[S] → L[T] assigning every n-ary P of L[S] injectively to a n-ary formula of L[T] and δ(x) is a formula of L[T] with one free variable different to all I(P), satisfying the following conditions:

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 4 / 17

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Definitions

Let L[S] and L[T] be languages of 1st order theories S and T with identity.

Definition (Relative Translation)

A relative translation f for L[S] to L[T] is given by a pair I, δ, where I : L[S] → L[T] assigning every n-ary P of L[S] injectively to a n-ary formula of L[T] and δ(x) is a formula of L[T] with one free variable different to all I(P), satisfying the following conditions: f (vn = vm) ˙ = (vn = vm) for all n, m ∈ N f (Pvi1...vin) ˙ = I(P)(vi1...vin) for all n-ary predicatsymbols P of L[S] f (¬ϕ) ˙ = ¬f (ϕ) and f (ϕ → ψ) ˙ = f (ϕ) → f (ψ) for all ϕ, ψ of L[S] f (∀vnϕ) ˙ = ∀vn(δ(vn) → f (ϕ)) for all ϕ of L[S] and all n ∈ N

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 4 / 17

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Definitions

Let L[S] and L[T] be languages of 1st order theories S and T with identity.

Definition (Relative Translation)

A relative translation f for L[S] to L[T] is given by a pair I, δ, where I : L[S] → L[T] assigning every n-ary P of L[S] injectively to a n-ary formula of L[T] and δ(x) is a formula of L[T] with one free variable different to all I(P), satisfying the following conditions: f (vn = vm) ˙ = (vn = vm) for all n, m ∈ N f (Pvi1...vin) ˙ = I(P)(vi1...vin) for all n-ary predicatsymbols P of L[S] f (¬ϕ) ˙ = ¬f (ϕ) and f (ϕ → ψ) ˙ = f (ϕ) → f (ψ) for all ϕ, ψ of L[S] f (∀vnϕ) ˙ = ∀vn(δ(vn) → f (ϕ)) for all ϕ of L[S] and all n ∈ N If f is a relative translation for L[S] to L[T], we write TrlL[S]

L[T](f ).

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 4 / 17

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Definitions

Definition (Relative Interpretation)

Let TrlL[S]

L[T](f ); we define f is a relative interpretation of S in T (S ≺f T)

and f is a relative faithful interpretation of S in T (S f T) as follows:

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

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Definitions

Definition (Relative Interpretation)

Let TrlL[S]

L[T](f ); we define f is a relative interpretation of S in T (S ≺f T)

and f is a relative faithful interpretation of S in T (S f T) as follows: S ≺f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇒ T ⊢ f (ϕ))

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

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Definitions

Definition (Relative Interpretation)

Let TrlL[S]

L[T](f ); we define f is a relative interpretation of S in T (S ≺f T)

and f is a relative faithful interpretation of S in T (S f T) as follows: S ≺f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇒ T ⊢ f (ϕ)) S f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇔ T ⊢ f (ϕ))

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

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Definitions

Definition (Relative Interpretation)

Let TrlL[S]

L[T](f ); we define f is a relative interpretation of S in T (S ≺f T)

and f is a relative faithful interpretation of S in T (S f T) as follows: S ≺f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇒ T ⊢ f (ϕ)) S f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇔ T ⊢ f (ϕ)) If there exists a f s.t S ≺f T (resp. S f T) we simply write S ≺ T (resp. S T) and speak of rel. interpretability (resp. rel. faithful interpretability) simpliciter.

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

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Definitions

Definition (Relative Interpretation)

Let TrlL[S]

L[T](f ); we define f is a relative interpretation of S in T (S ≺f T)

and f is a relative faithful interpretation of S in T (S f T) as follows: S ≺f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇒ T ⊢ f (ϕ)) S f T : ⇔ ∀ϕ(ϕ ∈ SentL[S] ⇒ (S ⊢ ϕ ⇔ T ⊢ f (ϕ)) If there exists a f s.t S ≺f T (resp. S f T) we simply write S ≺ T (resp. S T) and speak of rel. interpretability (resp. rel. faithful interpretability) simpliciter. If S ≺ T and T ≺ S (resp. S T and T S) we write S ∼ T (resp. S ≃ T) and speak of mutual (faithful) interpretability.

Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

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