Notes on the computational aspects of Kripkes theory of truth - PowerPoint PPT Presentation

Preliminaries Computational Aspects Some References Notes on the computational aspects of Kripke’s theory of truth Stanislav O. Speranski Associate Professor St. Petersburg State University Moscow 18.10.2017 S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Kripke’s Theory of Truth Consider the signature of Peano arithmetic and its expansion obtained by adding an extra unary predicate symbol T , viz. σ := { 0 , s , + , × , = } and σ T := σ ∪ { T } . Throughout this presentation the following assumptions are in force: the connective symbols are ¬ , ∧ and ∨ ; the quantifier symbols are ∀ and ∃ . We abbreviate ¬ ϕ ∨ ψ to ϕ → ψ , ( ϕ → ψ ) ∧ ( ψ → ϕ ) to ϕ ↔ ψ , etc. Let L and L T be the first-order languages of σ and σ T respectively. S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Here is some related notation: For := the collection of all L -formulas; Sen := the collection of all L -sentences; For T := the collection of all L T -formulas; Sen T := the collection of all L T -sentences . Assume some G¨ odel numbering # of L T has been chosen. Then we call A ⊆ N consistent iff there is no φ ∈ Sen T s.t. both # φ and # ¬ φ are in A . If A ⊆ N , we write � N , A � for the expansion of the standard model N of Peano arithmetic to σ T in which T is interpreted as A . S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References In his ‘Outline of a theory of truth’, Kripke used partial interpretations of T , i.e. pairs of the form S = � S + , S − � where S + and S − are disjoint subsets of N , resp. called the extension of S and the anti-extension of S . Henceforth we limit ourselves to partial interpretations of T with consis- tent extensions. A partial valuation for σ T is a mapping from Sen T to 0 , 1 � � a superset of 2 , 1 . By a valuation scheme we mean a function from partial interpretations to partial valuations. To begin with, let � sK and � wK be the orderings given by 1 1 0 � sK 2 � sK 1 and 2 � wK 0 � wK 1 . S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Define the strong Kleene valuation scheme V sK inductively as follows: for any closed L -terms t 1 and t 2 , � 1 if N | = t 1 = t 2 , V sK ( S ) ( t 1 = t 2 ) := 0 if N | = t 1 � = t 2 ; for every closed L -term t ,  if � N , S + � | 1 = T ( t ) ,   if � N , S − ∪ ( N \ # Sen T ) � | V sK ( S ) ( T ( t )) := 0 = T ( t ) , 1  otherwise;  2 V sK ( S ) ( ϕ ∧ φ ) := min � sK { V sK ( S ) ( ϕ ) , V sK ( S ) ( φ ) } ; V sK ( S ) ( ∀ x ϕ ( x )) := min � sK { V sK ( S ) ( ϕ ( t )) | t is a closed L -term } ; S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References V sK ( S ) ( ϕ ∨ φ ) := V sK ( S ) ( ¬ ( ¬ ϕ ∧ ¬ φ )); V sK ( S ) ( ∃ x ϕ ( x )) := V sK ( S ) ( ¬∀ x ¬ ϕ ( x )); V sK ( S ) ( ¬ ϕ ) := 1 − V sK ( S ) ( ϕ ). To get the weak Kleene valuation scheme V wK , simply replace � sK by � wK . Next we turn to so-called supervaluation schemes, each of which has the form  1 if for all A ⊆ N satisfying [*] , � N , A � | = ϕ,   V ( S ) ( ϕ ) := 0 if for all A ⊆ N satisfying [*] , � N , A � | = ¬ ϕ,  1 otherwise .  2 The best known such schemes are V SV , V VB , V FV and V MC , given by: S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References [*] = ‘ S + ⊆ A ’; V = V SV ⇐ ⇒ [*] = ‘ S + ⊆ A and A ∩ S − = ∅ ’; V = V VB ⇐ ⇒ [*] = ‘ S + ⊆ A and A is consistent’; V = V FV ⇐ ⇒ [*] = ‘ S + ⊆ A and A is cons. and complete’ . V = V MC ⇐ ⇒ Here the ‘completeness’ of A means that for each φ ∈ Sen T we have # φ ∈ A or # ¬ φ ∈ A . The last scheme emerges from Leitgeb’s ‘What truth depends on’ (although the definition presented below was stated explicitly by his PhD student Thomas Schindler). Say that ϕ ∈ Sen T depends on A ⊆ N iff for every B ⊆ N , � N , B � | = ϕ ⇐ ⇒ � N , B ∩ A � | = ϕ. S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Now define Leitgeb’s valuation scheme V L by if ϕ depends on S + ∪ S − and � N , S + � |  1 = ϕ,   if ϕ depends on S + ∪ S − and � N , S + � | V L ( S ) ( ϕ ) := 0 = ¬ ϕ, 1  otherwise .  2 It should be noted that each valuation scheme V induces a function J V from partial interpretations to partial interpretations, called the Kripke- jump operator for V , as follows: J V ( S ) + := { # ϕ | ϕ ∈ Sen T and V ( S ) ( ϕ ) = 1 } , J V ( S ) − := { # ϕ | ϕ ∈ Sen T and V ( S ) ( ϕ ) = 0 } ∪ ∪ { n ∈ N | n �∈ # Sen T } . S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References In turn J V generates a transfinite sequence indexed by ordinals:  S if α = 0 ,   � �  J β J α V ( S ) if α = β + 1 , V ( S ) := J V + , �  − � β<α J β β<α J β  � � V ( S ) V ( S ) if α ∈ L-Ord .  V ( ∅ , ∅ ) + — these sets constitute We shall often write T α V instead of J α the truth hierarchy for V . Moreover Kripke dealt with monotone schemes, i.e. those which satisfy the condition that for any partial interpretations S 1 and S 2 , S + 1 ⊆ S + & S − 1 ⊆ S − = ⇒ 2 2 J V ( S 1 ) + ⊆ J V ( S 2 ) + & J V ( S ) − ⊆ J V ( S ) − . = ⇒ S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Observation (Kripke) For every monotone valuation scheme V there exists an ordinal α s.t. V = T α +1 T α — yielding the least fixed point of J V . V It is easy to verify that each V ∈ { V sK , V wK , V SV , V VB , V FV , V MC , V L } is monotone and furthermore has the following properties: if J V ( S ) = S , then V ( S ) ( T ( � ϕ � )) = V ( S ) ( ϕ ); V ( S ) − iff # ¬ ϕ ∈ J α V ( S ) + ; # ϕ ∈ J α V ( S ) + iff # ¬ ϕ ∈ J α # ϕ ∈ J α V ( S ) − ; J V turns out to be a ‘Π 1 1 -operator’ — so by a well-known theorem V = T α +1 of Spector, T α � ω CK � already for some α ∈ C-Ord ∪ . 1 V S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Kleene’s O Remember that Kleene’s system of notation for C-Ord consists of: a special partial function ν O from N onto C-Ord; an appropriate ordering relation < O on dom ( ν O ) — which mimics the usual ordering relation on C-Ord. Call n ∈ N a notation for α ∈ C-Ord iff ν O ( n ) = α . To simplify the sta- tements I often write n ∈ O instead of n ∈ dom ( ν O ). Folklore dom ( ν O ) is Π 1 1 -complete. S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries Kripke’s Theory of Truth Computational Aspects Kleene’s O Some References Fix one’s favorite universal partial computable (two-place) function U . Folklore There exists a computable function f such that for every n ∈ O , � � { k ∈ N | k < O n } = dom U f ( n ) . Folklore (Effective Transfinite Recursion) Suppose f is a computable function such that for any e ∈ N and n ∈ O , � � { k ∈ N | k < O n } ⊆ dom ( U e ) = ⇒ n ∈ dom U f ( e ) . Then there is a c ∈ N for which U f ( c ) = U c , and dom ( ν O ) ⊆ dom ( U c ) . S. O. Speranski On the computational aspects of Kripke’s theory of truth

Preliminaries About Least Fixed-Points Computational Aspects Some Strengthenings Some References About Least Fixed-Points Let us call a valuation scheme V ordinary iff for any α ∈ Ord, χ ∈ Sen , ψ ∈ Sen T and ϕ ( x ) ∈ For T the following conditions hold: T α V ⊆ T α +1 ; 1 V χ ∈ T α V iff α � = 0 and N | = χ ; 2 V iff T ( � ψ � ) ∈ T α +1 ψ ∈ T α ; 3 V ∀ x ϕ ( x ) ∈ T α +1 iff { ϕ ( n ) | n ∈ N } ⊆ T α +1 ; 4 V V χ ∧ ψ ∈ T α = χ and ψ ∈ T α V iff N | V ; 5 if χ ∨ ψ ∈ T α = ¬ χ , then ψ ∈ T α V and N | V ; 6 = χ and α � = 0, then χ ∨ ψ ∈ T α if N | V . 7 In effect, except for V wK , all the schemes considered above are ordinary. S. O. Speranski On the computational aspects of Kripke’s theory of truth