12th Meeting on Mathematics of Language. 6, September, 2011
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1 A Co-inductive Collaborate Research Team for Verification, National Institute of Advanced Industrial Science and Technology, Japan 12th Meeting on Mathematics of Language 6, September, / 17
2 Outline Objective: (computational) analysis of truth in fuzzy logics The problem: A fuzzy truth theory, PAŁTr 2 has an unexpected properties: A formalized version of truth fails, It is due to the fact that the formalized commutation scheme (FC) is contradictory on PAŁTr 2. We observe that it can be expressed in terms of co-induction They are caused by the fact that the truth predicate enables to define formulae co-inductively, the unprovability of FC is caused by the guarded co-recursion. 2 / 17
3 The motivations for linguists (1) Fuzzy logics are used to represent graduality in semantics of natural, attributed to some objects more than to others humid, hot, green, expensive, difficult, expressed by adjectives that is comparative reducible to a real-valued quantity (e.g. humid in %) (2) The of PAŁTr 2 is a co-inductive, allows to define sentences of infinite length virtually by using the truth predicate, co-induction plays a key role in computer sciences to represent non-terminate automata [MT91]: it is ubiquitous, the paradoxical sentence is like a non-terminate automaton! 3 / 17
4 The objective: a fuzzy logic Łukasiewicz infinite-valued predicate logic Ł: Truth values are real numbers in [0, 1], ϕ 0 ϕ 1 = min{1, 1 ϕ 0 + ϕ 1 }, = 0, A A, etc. ( x)ϕ(x) = inf{ ϕ(a) M : a M }. Ł is a sublogic of classical logic. 4 / 17
5 The objective: truth degrees Its meaning is often expressed by truth degrees: The basic idea: a formula is true to some degree, we can construct truth degrees, degrees of all sentences in any algebra from a viewpoint of metatheory: for any sentence A, B, A B is true, i.e. A B if and only if A B = 1 often think that they represent degrees of truthhood : They coincide: truth of A implies truth of B : A B A B A B = 1 5 / 17
6 Our framework PAŁTr 2 Axiomatic truth theory PAŁTr 2 [HPS00] over Ł with unary predicate Tr(x), T-schemata (where ϕ is the Gödel code of ϕ) ϕ Tr( ϕ ) all axioms of classical PA, the induction scheme (extended) The unrestricted form of T-schemata is not contradictory in PAŁTr 2. The liar sentence, Λ Tr( Λ ), dose not imply a contradiction in Ł: Λ = 0.5, 6 / 17
7 Formzlization: a formalized truth in PAŁTr 2 Some philosophers, as deflationists, believe T-schemata are the foundational concept of truth, Can we formalize in PAŁTr 2? degree theoretic ordering: ( x, y)(form(x)&form(y) [x y (Tr(x y))], i.e. the standard calculation way of truth degrees: degrees of truthhood: ( x, y)(form(x)&form(y) [x y (Tr(x) Tr(y))] [Fl08], The formalized truth identity with : ( x, y)(form(x)&form(y) [x y x y]) 7 / 17
8 However, the formalized truth badly fails: PAŁTr 2 proves ( x, y)(form(x)&form(y) [x y x y]) It is due to the following facts: the formalized commutativity (FC) implies a contradiction [HPS00]: ( x, y)(form(x)&form(y) [Tr(x y) (Tr(x) Tr(y))]) Quite contrary, for any formula ϕ, ψ, PAŁTr 2 proves the following: ϕ ψ ϕ ψ 8 / 17
9 The paradoxical formula formal definition: λ ( x)tr( x λ ) where (n + 1) A A (n A). It s easy to see n A = min{(n + 1) A, 1}, n A (n + 1) A, intuitive meaning: an infinite stream λ λ (λ (λ (λ ))) 9 / 17
10 The failure of FC PAŁTr 2 implies λ [R93]. The mathematical induction and FC imply λ. Initial case: Tr( 0 λ ) = 1. Induction step: assume Tr( n λ ) = 1. λ implies Tr( λ ) by FC, and the induction hypothesis implies Tr( λ )& Tr( n λ ) = 1, this is equivalent to (Tr( λ ) Tr( n λ )), FC implies Tr( λ n λ ), this is equivalent to Tr( (n + 1) λ ). Tr( λ ) Tr( n λ ) Tr( (n + 1) λ ) implies 10 / 17
11 λ is finite, but it generates (or unfolds) the infinite sentence inside the truth predicate virtually [Y11]: λ ( x)tr( λ (x λ) ) ( y)tr( λ (λ (y λ)) ) Tr( λ (λ (λ (λ )) ) The keyword: co-induction A typical example of co-inductive definition: for any set A, the infinite stream of A is of the type A, γ : A (A A ). Intuitively speaking, this definition describes that, for any a 0, a 1, A, γ( a 0, a 1, ) = (a 0, a 1, ) (A A ) γ gets rid of the first element a 0 of the infinite stream (but a 1, is still infinite). 11 / 17
12 Truth predicate expands definability of operations of fromulae. Formally speaking, is defined (as arithmetical function) as follows: (a) f(0, x) = x, (b) f(n + 1, x) = x f(n, x). (c) n A is just an abbreviation of Tr( f(n, A )). Tr interprets arithmetical operations on Godel codes to real operations on formulae. Because of the existence of full T-scheme, we can identify Godel codes as formulae themselves. So recursive functions over Godel codes are translated to infinite operations on formulae. 12 / 17
13 What is a cause of the problem? Let us remember: the mathematical induction proves FC implies λ. PAŁTr 2 proves (Tr( λ ) Tr( x λ )), FC proves Tr( λ x λ ) What is a difference? Unrestricted forms of co-recursion (recursive definition over co-inductive objects) is often contradictory to induction: we can define functions whose computation does not terminate in finite many steps, only a restricted form, the guarded co-recursion, is allowed in many case. 13 / 17
14 (Rough) Example for any recursive function f, map : (A B) A B is defined as map f x, x 0, = f(x) ( map f x 0, x 1, ) Here recursive call of map only appears inside, The inside and outside of are essentially different. PAŁTr 2 case Tr( d + 1 λ ) Tr( Tr( λ ) (d λ) ) holds, all λs are guarded because all recursive call of λ are inside the same Tr. however Tr( d + 1 λ ) Tr( λ ) Tr( d λ ) might not hold, some of the recursive calls of λ are not inside the same Tr, therefore they are not guarded. when d is a nonstandard number. 14 / 17
15 There are two ways to define truth degrees: Degrees of truthhood: ( x, y)x y Tr(x) Tr(y) Degree theoretic ordering: ( x, y)x y Tr(x y) But they do not coincide in general: and coincide for any formulae A, B, They are different for non-standard Godel numbers: Degrees of truthhood are still gradual, The graduality of Truth degrees is violated because it is possible to define the limit, λ, of the infinitary operation by using co-recursion: λ λ ( λ ( λ ( λ )) The truth predicate allows us to define formulae co-inductively. 15 / 17
16 Future tasks Other paradoxes? Other logics? Other concepts? Natural s? 16 / 17
17 Reference Hartry Field. Saving Truth From Paradox Oxford (2008) Petr Hájek, Jeff B. Paris, John C. Shepherdson. The Liar Paradox and Fuzzy Logic Journal of Symbolic Logic, 65(1) (2000) Robin Milner, Mads Tofte. in relational semantics Theoretical computer science 87 (1991) Greg Restall Arithmetic and Truth in Łukasiewicz s Infinitely Valued Logic Logique et Analyse 36 (1993) Yablo-like paradoxes and co-induction Lecture Note in Computer Science 6797, springer. 17 / 17
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