Non-classical circular definitions Shawn Standefer University of - PowerPoint PPT Presentation

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Non-classical circular definitions Shawn Standefer University of Melbourne Frontiers of Non-Classicality January 27, 2016

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Plan Cover a little historical context and motivation for the study of circular definitions Sketch some features of the Strong Kleene fixed-point theory Sketch some features of the supervaluation fixed-point theory Go over some of the features of a particular class of definitions

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Theories of truth – fixed-point Saul Kripke, and independently Robert Martin and Peter Woodruff, came up with fixed-point theories of truth, similar to the work of Paul Gilmore and Ross Brady. The idea is that the semantic value of the truth predicate for a language is a fixed-point of an operation on possible 3-valued interpretations. Given certain interpretations as starting points, one can view the operation as building up fixed-points inductively.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Theories of truth – revision Anil Gupta and, independently, Hans Herzberger created revision theories for truth Use sequences of 2-valued interpretations of the truth predicate Sequences are generated by an operation on interpretations determined by: ‘ A ’ is true iff A , not understood as the material biconditional. The interpretations need not reach fixed-points, but certain sentences will stabilize as 1 or 0.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References From truth to definitions Tarski biconditionals: ‘ A ’ is true iff A Gupta and Belnap say that the Tarski biconditionals together provide a circular definition of truth. Generalize to circular definitions

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Circular definitions Given a language L , expand the language with new predicate letters G i , each of which receives a definitional clause, to obtain L + . Permit any set of interdependent definitions. G 1 ( x 1 ) = Df A G 1 ( x 1 ) G 2 ( x 2 ) = Df A G 2 ( x 2 ) . . . G k ( x k ) = Df A G k ( x k ) . . . A G i is any formula of the language L + .

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Examples G : Gx = Df Gx ∨ ∼ Gx L : Lx = Df ∼ Lx . J : Jx = Df DLINB ( < ) & ∀ y ( y < x ⊃ Jy ) & ∼∀ yJy

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Revision theory of definitions General circular definitions provide new options for analyzing interdependent concepts. One application: an alternative account of rationality in game theory (Chapuis (2003), Gupta (2000), Bruni). This has all been done in the classical scheme, so there are questions about circular definitions in other settings.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Another way in Moschovakis (1994, 2006) proposes understanding Frege’s distinction between sense and reference as algorithm and value, respectively. He formalizes this idea using languages with explicit self-reference, introducing new predicates that are defined via algorithms provided by arbitrary formulas of the expanded language. His concern is with computation, so he focuses on the Strong Kleene scheme. Although Moschovakis defines algorithms in terms of fixed-points, his motivations for this proposal rely on the intuition of stepping through an iteration.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Another way in This presentation follows Gupta and Belnap’s approach, rather than Moschovakis’s. I will focus on Strong Kleene definitions for most of the talk, although there are no philosophical barriers to looking at supervaluation (or LP, or fuzzy, or. . . ) definitions. The goal is to better understand the logic of non-classical definitions so as to compare it to the classical revision theory of definitions.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Formalism Base language L interpreted via a classical ground model M (= � D , I � ). Expand language to L + with new predicates defined via the set D A hypothesis h is a function from defined predicates to functions from tuples from D to truth values – h : D �→ ( D n �→ { 1 , 0 , 1 2 } ). Hypotheses interpret defined predicates.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Semantics Model M + h is just like M except that h interprets the defined predicates. V M + h ( Gt ) = h ( G )( I ( t )) V M + h ( Ft ) = I ( F )( I ( t ))

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Strong Kleene scheme The semantic values are linearly ordered by the logical ordering: 0 ≤ L 1 2 ≤ L 1 1 ∼ ∨ 1 0 2 1 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 0 1 0 1 0 2 V M + h ( ∀ xA ( x ))) = min ( { V M + h ( A ( d )) : d ∈ D } )

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Ordering The semantic values are partially ordered by the information ordering: 1 2 ≤ i 1 and 1 2 ≤ i 0. Use this to define a partial order on hypotheses h � h ′ iff for all predicates G in D and all appropriate length tuples d , h ( G )( d ) ≤ i h ′ ( G )( d ).

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Definitions A set of definitions D yields a jump (or revision) operator κ D , M . κ D , M ( h )( G )( d ) = Val M + h ( A G ( d )) The Strong Kleene scheme is monotonic κ D , M is monotonic, i.e. h � h ′ ⇒ κ D , M ( h ) � κ D , M ( h ′ )

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Definitions A hypothesis h is sound iff h � κ D , M ( h ) A hypothesis f is a fixed-point iff f = κ D , M ( f ). Given monotonicity, iterating κ D , M , possibly transfinitely, on a sound h will yield a fixed-point, f with h � f . In particular, iterating κ D , M on the � -minimal hypothesis h 0 will yield the minimal or least fixed-point. Use fixed-points to interpret defined predicates.

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Entailment For a given language L and set of definitions D A 1 , . . . , A n entails B 1 , . . . , B m in M on D = SK , M ( A 1 , . . . , A n | B 1 , . . . , B m ) iff for all fixed-points f , if for all D i ≤ n , V M + f ( A i ) = 1, then for some i ≤ m , V M + f ( B i ) = 1. = SK A 1 , . . . , A n entails B 1 , . . . , B m on D ( A 1 , . . . , A n | B 1 , . . . , B m ) D iff for all classical ground models M , = SK , M A 1 , . . . , A n | B 1 , . . . , B m . D

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Features = SK B 1 , . . . , B m iff A ′ 1 , . . . , A ′ = SK B ′ 1 , . . . , B ′ A 1 , . . . , A n | n | m , where D D ′ indicates possibly replacing occurrences of Gt with A G ( t ), or conversely. = SK We can axiomatize | D , building on Kremer (1988).

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Sequents: Structural rules Γ ⊢ SK ∆ Axioms: D ( ⊢ K ) Γ ⊢ SK ∆ , A D A ⊢ SK A D ∼ A , A ⊢ SK D A , A , Γ ⊢ SK ∆ D ⊢ SK ( W ⊢ ) B , ∼ B A , Γ ⊢ SK D ∆ D B is D -free Structural rules: Γ ⊢ SK Γ ⊢ SK ∆ ∆ , A , A D D ( K ⊢ ) ( ⊢ W ) A , Γ ⊢ SK Γ ⊢ SK ∆ ∆ , A D D

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Sequents: Connectives Γ ⊢ SK ∼ B , Γ ⊢ SK ∆ , A ∆ D D ( ⊢∨ ) ( ∼ ∨ ⊢ ) Γ ⊢ SK ∼ ( A ∨ B ) , Γ ⊢ SK ∆ , A ∨ B ∆ D D Γ ⊢ SK ∆ , B Γ ⊢ SK Γ ⊢ SK ∆ , ∼ A ∆ , ∼ B D D D ( ⊢∨ ) ( ⊢∼∨ Γ ⊢ SK Γ ⊢ SK ∆ , A ∨ B ∆ , ∼ ( A ∨ B ) D D A , Γ ⊢ SK B , Γ ⊢ SK A , Γ ⊢ SK ∆ ∆ ∆ D D D ( ∨⊢ ) ( ∼∼⊢ ) A ∨ B , Γ ⊢ SK ∼∼ A , Γ ⊢ SK ∆ ∆ D D ∼ A , Γ ⊢ SK Γ ⊢ SK ∆ ∆ , A D D ( ∼ ∨ ⊢ ) ( ⊢∼∼ ) ∼ ( A ∨ B ) , Γ ⊢ SK Γ ⊢ SK ∆ ∆ , ∼∼ A D D

Bit of background Fixed-point theories of definitions Supervaluations Definitions References Sequents: Quantifiers A [ t / x ] , Γ ⊢ SK ∼ A [ y / x ] , Γ ⊢ SK ∆ ∆ D D ( ∀⊢ ) ( ∼∀⊢ ) ∀ xA , Γ ⊢ SK ∼∀ xA , Γ ⊢ SK ∆ ∆ D D Γ ⊢ SK Γ ⊢ SK ∆ , A [ y / x ] ∆ , ∼ A [ t / x ] D D ( ⊢∀ ) ( ⊢∼∀ ) Γ ⊢ SK Γ ⊢ SK ∆ , ∀ xA ∆ , ∼∀ xA D D In ( ⊢∀ ) and ( ∼∀⊢ ), the variable y cannot occur freely in the conclusion sequents.