Reflecting on truth in a partial setting Martin Fischer MCMP LMU Bristol-M¨ unchen Conference on Truth and Rationality 10. 6. 2016 Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 1 / 36
Overview The study of reflection principles are important in the arithmetical setting. Also for theories of truth the investigation of reflection principles is important and fruitful. What about reflection principles in a partial setting? What about the connection between reflection and PKF? Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 2 / 36
Content Background 1 Axiomatizing Kripke N -Categoricity Infinitary proof systems Reflection 2 From the ω -rule to reflection From Tarski biconditionals to KF Reflecting on truth in a partial setting 3 Partial logic Recovering PKF Induction Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 3 / 36
Background Axiomatizing Kripke Kripke models Kripke: Fixed-point construction for different evaluation schemes e . monotone operators Γ e . Fixed-points Γ e ( S ) = S for S ⊆ N . Focus: strong Kleene, e = sk . The minimal fixed-point for strong Kleene I sk . Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 4 / 36
Background Axiomatizing Kripke Axiomatizing Kripke Axiomatizations: KF (Feferman) The problem of external and internal logic. IKF (Reinhardt) ( { A ∈ L T | KF ⊢ T ( � A � ) } ) The problem of natural axiomatization. PKF (Halbach/Horsten) In what sense are these axiomatizations and which one is preferable? Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 5 / 36
Background N -Categoricity N -Categoricity Suggestion: N -categoricity. Fix the interpretation of the arithmetical part with the standard model N . Σ is N -categorical for a set of models M iff ( N , S ) | = Σ ⇔ S ∈ M For the minimal fixed-point: ( N , S ) | = Σ ⇔ S = I sk For arbitrary fixed-points: ( N , S ) | = Σ ⇔ S = Γ sk ( S ) Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 6 / 36
Background N -Categoricity N -Categoricity The minimal fixed-point is Π 1 1 -complete (Kripke, Burgess). There is no N -categorical axiomatization of the minimal fixed-point. KF is an N -categorical axiomatization of arbitrary fixed-points. (Feferman) TFB is an N -categorical axiomatization of arbitrary fixed-points. (Leigh) IKF is not N -categorical axiomatization of arbitrary fixed-points. Conclusion: KF is at best an axiomatization of arbitrary fixed-points and N -categoricity cannot be the only criterion. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 7 / 36
Background N -Categoricity N -Categoricity and partiality The set of derivable sequents of PKF is an N -categorical axiomatization of arbitrary fixed-points. The set of theorems of PKF, i.e. sequents of the form ⇒ A , is not N -categorical axiomatization of arbitrary fixed-points. The set of truth sequents T ( � A � ) ⇒ A , A ⇒ T ( � A � ) is an N -categorical axiomatization of arbitrary fixed-points. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 8 / 36
Background Infinitary proof systems Infinitary proof systems Infinitary proof systems allow for characterizations of the minimal fixed-points. Cantini has an infinitary proof system (sequent system with ω -rule) characterizing the minimal fixed-point of supervaluation. Welch gametheoretic characterization. Meadows infinitary tableaux. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 9 / 36
Background Infinitary proof systems Infinitary proof system for strong Kleene Example SK ∞ a Tait system: Initial sequents ⇒ A (for true atomic arithmetical sentences) ⇒ A ⇒ ¬ A ⇒ Γ , T ( � A � ) ⇒ Γ , ¬ T ( � A � ) ... A ( n ) ... (for all n ∈ N ) ω -rule ∀ xA ( x ) Then SK ∞ ⊢ A ⇔ # A ∈ I sk Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 10 / 36
Background Infinitary proof systems Embeddings into infinitary proof systems Similar to the Gentzen-Sch¨ utte method we can look at embeddings into the infinitary proof systems. KF cannot be directly embedded. An embedding of the theorems of PKF into SK ∞ is possible ◮ if PKF ⊢⇒ A , then # A ∈ Γ ω ω (Cantini, Halbach/Horsten). ◮ for the language of truth we only have transfinite induction up to ω ω in PKF. IKF is contained in I sk ◮ if IKF ⊢ A , then # A ∈ Γ ǫ 0 (Cantini). ◮ for the language of truth we have transfinite induction up to ǫ 0 in KF. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 11 / 36
Reflection From the ω -rule to reflection Uniform reflection as a finitary ω -rule ∀ x Pr Σ ( � A ˙ x � ) (RFN R Σ ) ∀ xA ( x ) (RFN Σ ) ∀ x (Pr Σ ( � A ˙ x � ) → A ( x )) . Hilbert 1931. Shoenfield constructivized version of the ω -rule. Feferman 1962 showed the equivalence. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 12 / 36
Reflection From the ω -rule to reflection The strength of uniform reflection For an axiomatizable theory Σ we use R(Σ) := EA T + RFN Σ . TB 0 is EA T + Tarski biconditionals for sentences of L A . UTB 0 is EA T + uniform Tarski biconditionals for formulas of L A . TFB 0 is EA T + truth and falsity biconditionals for sentences of L P , i.e. the language of we get by adding F as the dual for T and allow only positive occurrences of T and F . T ( � A � ) ↔ A & F ( � A � ) ↔ A UTFB 0 is EA T + uniform truth and falsity biconditionals for formulas of L P . Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 13 / 36
Reflection From Tarski biconditionals to KF Truth and Reflection Reflecting on Tarski biconditionals gives uniform Tarski biconditionals. Lemma (Horsten, Leigh) UTB 0 ⊆ R(TB 0 ) . Reflecting on typefree truth and falsity biconditionals gives uniform typefree truth and falsity biconditionals. Lemma (Horsten, Leigh) UTFB 0 ⊆ R(TFB 0 ) . Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 14 / 36
Reflection From Tarski biconditionals to KF Truth and Reflection Reflecting on uniform Tarski biconditionals gives the compositional axioms. Lemma (Halbach) CT 0 ⊆ R(UTB 0 ) . Reflecting on uniform truth and falsity biconditionals gives the compositional axioms of KF. Lemma (Horsten, Leigh) KF ⊆ R(UTFB 0 ) . Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 15 / 36
Reflecting on truth in a partial setting Partial logic Partial logic The logic is four valued. Gaps and gluts. Logical consequence for sequents: ◮ Truth preservation ◮ Falsity antipreservation Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 16 / 36
Reflecting on truth in a partial setting Partial logic Basic For negation we have contraposition Γ ⇒ ∆ ¬ ∆ ⇒ ¬ Γ but not A , Γ ⇒ ∆ Γ ⇒ ∆ , A Γ ⇒ ∆ , ¬ A ¬ A , Γ ⇒ ∆ We assume as background an arithmetical theory BASIC formulated in L T : EA T formulated in a sequent version of partial logic along the lines of Halbach 2014. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 17 / 36
Reflecting on truth in a partial setting Partial logic Minimal truth TS 0 TS 0 is obtained by extending BASIC with the initial sequents T 1 T ( � A � ) ⇒ A A ⇒ T ( � A � ) T 2 Simplicity. No need for restriction of the language. Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 18 / 36
Reflecting on truth in a partial setting Partial logic Reflection as a rule Assume some coding of finite sets of formulas [Γ], then [Γ˙ x ] denotes the result of substituting in Γ the x -th numeral for x . x ] ⇒ [Γ˙ . [∆˙ x ] denotes the sequent Γ( x ) ⇒ ∆( x ) with the possible free variable x and the dots indicate as usual the use of the sub and num function. Let Σ be an axiomatizable theory, then R(Σ) = EA T + RFN R Σ . Pr Σ ([Γ˙ x ] ⇒ . [∆˙ x ]) (RFN R Σ ) Γ( x ) ⇒ ∆( x ) Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 19 / 36
Reflecting on truth in a partial setting Recovering PKF From TS 0 to UTS 0 R(TS 0 ) ⊢ (i) A ( x ) ⇒ T ( � A ˙ x � ); (ii) T ( � A ˙ x � ) ⇒ A ( x ). Argument: For all formulas A ( x ) and for all n ∈ N : TS 0 ⊢ A ( n ) ⇒ T ( � A ( n ) � ) . As this is uniform we get in the formalization EA T ⊢⇒ Pr TS 0 ([ A ˙ x ] ⇒ . [ T ( � A � )˙ x ]) . With reflection we get R(TS 0 ) ⊢ A ( x ) ⇒ T ( � A ˙ x � ) . Martin Fischer (MCMP LMU) Reflecting on truth in a partial setting Bristol-M¨ unchen 20 / 36
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