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

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

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

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

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

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

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

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

Overview Relative Interpretation 1 Definitions Examples Basic Principles Preservation of Meaning 2 Meaning and Interpretation Preservation of Meaning A Counterexample 3 Expressing Inconsistency Feferman’s Theorem A Counterexample Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 3 / 17

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

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

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 ( v n = v m ) ˙ = ( v n = v m ) for all n , m ∈ N f ( Pv i 1 ... v i n ) ˙ = I ( P )( v i 1 ... v i n ) 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 ( ∀ v n ϕ ) ˙ = ∀ v n ( δ ( v n ) → 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

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 ( v n = v m ) ˙ = ( v n = v m ) for all n , m ∈ N f ( Pv i 1 ... v i n ) ˙ = I ( P )( v i 1 ... v i n ) 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 ( ∀ v n ϕ ) ˙ = ∀ v n ( δ ( v n ) → 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 Trl L [ S ] L [ T ] ( f ) . Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 4 / 17

Definitions Definition (Relative Interpretation) Let Trl L [ 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

Definitions Definition (Relative Interpretation) Let Trl L [ 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 ∈ Sent L [ S ] ⇒ ( S ⊢ ϕ ⇒ T ⊢ f ( ϕ )) Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

Definitions Definition (Relative Interpretation) Let Trl L [ 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 ∈ Sent L [ S ] ⇒ ( S ⊢ ϕ ⇒ T ⊢ f ( ϕ )) S � f T : ⇔ ∀ ϕ ( ϕ ∈ Sent L [ S ] ⇒ ( S ⊢ ϕ ⇔ T ⊢ f ( ϕ )) Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 5 / 17

Definitions Definition (Relative Interpretation) Let Trl L [ 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 ∈ Sent L [ S ] ⇒ ( S ⊢ ϕ ⇒ T ⊢ f ( ϕ )) S � f T : ⇔ ∀ ϕ ( ϕ ∈ Sent L [ 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

Definitions Definition (Relative Interpretation) Let Trl L [ 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 ∈ Sent L [ S ] ⇒ ( S ⊢ ϕ ⇒ T ⊢ f ( ϕ )) S � f T : ⇔ ∀ ϕ ( ϕ ∈ Sent L [ 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

Examples PA ≺ ZF Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Hyperbolic Geometry H ≺ RCF Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Hyperbolic Geometry H ≺ RCF Robinson Arithmetic Q ≺ First-order Group Theory Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Hyperbolic Geometry H ≺ RCF Robinson Arithmetic Q ≺ First-order Group Theory PA + ¬ Con pa ≺ PA Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Hyperbolic Geometry H ≺ RCF Robinson Arithmetic Q ≺ First-order Group Theory PA + ¬ Con pa ≺ PA , but PA + Con pa �≺ PA Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17

Examples PA ≺ ZF Euclidian Geometry P ≺ RCF Hyperbolic Geometry H ≺ RCF Robinson Arithmetic Q ≺ First-order Group Theory PA + ¬ Con pa ≺ PA , but PA + Con pa �≺ PA ZF + GCH + AC ∼ ZF Mirko Engler (HU Berlin & UNL) Meaning & Interpretation Prague, May 2, 2018 6 / 17