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Ordinal foundation for Łukasiewicz semantics 
Rossella Marrano 
Scuola Normale Superiore, Pisa 
LATD 2014, Vienna, 17 July 2014 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 1 / 16
Motivation 
Degrees of truth as real numbers 
We shall assume that the truth degrees are linearly ordered, with 1 
as maximum and 0 as minimum. Thus truth degrees will be coded 
by (some) reals. [. . . ] We shall always take the set [0; 1] with its 
natural (standard) linear order. (Hájek, 1998) 
Artificial precision 
I arbitrariness of the choice 
how can we justify the choice of the truth value 0.24 over 0.23? 
I implausibility of the interpretation 
what does it mean for a sentence to be 1= true? 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 2 / 16
Motivation (cont.) 
Unfolding the objection 
% the logic assigns a certain real number to sentences 
% how to measure exactly the value 
 unique, exact real number as truth-value 
 measurability in principle 
Diagnosis 
I point-wise valuation 
I numerical (cardinal) assignment 
Our proposal: ordinal foundation 
Pairwise valuations (‘being more or less true’) with no intensity 
Representation theorems Can infinite-valued valuations be proved to arise 
from truth-comparisons? 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 3 / 16
Łukasiewicz infinite-valued logic 
I L = fp1; p2; : : : g 
I :, !, ? 
I SL 
I `Ł 
(Ł1)  ! ( ! ) 
(Ł2) ( ! ) ! (( ! ) ! ( ! )) 
(Ł3) (: ! :) ! ( ! ) 
(Ł4) (( ! ) ! ) ! (( ! ) ! ) 
(MP) 
Standard truth-value semantics v : SL ! [0; 1] 
1. v(?) = 0. 
2. v(:) = 1  v() 
 
3. v( ! ) = 
1; if v()  v(); 
1  v() + v(); otherwise. 
Order-based semantics   SL  SL 
I    ()def    and   : 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 4 / 16
Structural constraints 
(A.1)  is transitive 
(A.2) ?   
Soundness 
(A.3) `  =)    
Truth-condition for the implication 
(A.4)    ()  !    
Monotonicity constraints 
(A.5) 1  2; 1  2 =) 1 ! 1  2 ! 2 
(A.6)    =) :  : 
Theorem 
If  satisfies axioms (A.1)–(A.6) then there exists a Łukasiewicz valuation 
v : SL ! [0; 1] such that for all ;  2 SL: 
   =) v()  v(): 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 5 / 16
Some consequences of the axioms 
1.  is reflexive 
2.    
Define    := : !  and
:= :(:  :). If axiom (A.6) is satisfied, 
then axiom (A.5) is equivalent to the following: 
(A.50) 1  2; 1  2 =) 1
1  2
2 
(A.500) 1  2; 1  2 =) 1  1  2  2 
1. `  !  =)    
2. `  $  =)    
1.  is an equivalence relation 
2. 1  1; 2  2 =) 1  2  1  2 
3.    =) :  : 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 6 / 16
An algebra over (SL;) 
I SL== f [] j  2 SL g 
I [] = f  j    g 
 
? := [?] := ? 
I : 
I 
[] := [:] 
I [] 
 
[] := [  ] 
I []  [] () 9i 2 []; i 2 [] i  i 
I []  [] () [] 
! 
[] = [] 
: 
and 
are well defined. 
The two orders,  and , coincide. 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 7 / 16
Relation to the Lindenbaum algebra 
   ()def `  $  
(SL;) (SL 
 ;:;; 0;) 
(SL 
 ; 
: 
; 
 
; 
 
?;) 
q 
q 
f([]) = [] 
Lemma 
f(q()) = q(). f is well defined, onto and it is a homomorphism. 
Lemma 
(SL=; 
: 
; 
 
; 
 
?) is a non-trivial MV-algebra. 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 8 / 16
Theorem (Cignoli et al. 2000) 
If M is a non-trivial MV-algebra then there exists at least one homomorphism: 
m: M ! [0; 1]MV : 
q m 
(SL;) (SL=; 
: 
; 
 
; 
 
?;) ([0; 1];:;; 0;) 
V 
I There exists V : SL ! [0; 1]. 
I V is a Łukasiewicz valuation. 
I V preserves , namely:    =) V()  V(): 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 9 / 16
Corollaries 
(A.C)    or    
(A.A) if    and   ? then 9n |  {z  } 
n 
  
Corollary 
If  satisfies axioms (A.1)–(A.6) and (A.C) then there exists a unique 
valuation representing . 
Corollary 
If  satisfies axioms (A.1)–(A.6), (A.C) and (A.A) then there exists a unique 
Łukasiewicz valuation v : SL ! [0; 1] such that for all ;  2 SL: 
   () v()  v(): 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 10 / 16
Completeness 
I j=  ()def for all  satisfying (A.1)–(A.6)    
I  j=  ()def for all  satisfying (A.1)–(A.6) if 8 2    
 then    
Soundness 
It easily follows from the axioms. 
Completeness 
8 2 SL j=  =) ` : 
I j=  =) `  [Rose-Rosser, 1958] 
I j=  =) j=  
Strong completeness 
8 2 SL; 8  SL  j=  =)  ` : 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 11 / 16
Desirability of the axioms 
(A.1)  is transitive 
(A.2) ?   
(A.3) `  =)    
(A.4)    ()  !    
(A.5) 1  2; 1  2 =) 1 ! 1  2 ! 2 
(A.6)    =) :  : 
(A.C)    or    
(A.A) if    and   ? then 9n |  {z  } 
n 
  
The axiom (A.3): `  =)    
i. abbreviation 
ii. soundness 
iii. non-justificative aim 
iv. generalization 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 12 / 16
Generalization (future work) 
Why Łukasiewicz? 
The objection does not apply with the same strength to any logic with a 
[0; 1]-valued semantics 
Possible generalizations 
Remark 
Let v be a Łukasiewicz (Gödel, Product logic) valuation. The order defined by 
 v  ()def v()  v() 
satisfies axioms (A.C), (A.1)–(A.6) where ` is the Łukasiewicz (Gödel, 
Product logic) deducibility relation. 
I sufficient conditions 
I a general case 
Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 13 / 16

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RossellaMarrano_LATD2014

  • 1. Ordinal foundation for Łukasiewicz semantics Rossella Marrano Scuola Normale Superiore, Pisa LATD 2014, Vienna, 17 July 2014 Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 1 / 16
  • 2. Motivation Degrees of truth as real numbers We shall assume that the truth degrees are linearly ordered, with 1 as maximum and 0 as minimum. Thus truth degrees will be coded by (some) reals. [. . . ] We shall always take the set [0; 1] with its natural (standard) linear order. (Hájek, 1998) Artificial precision I arbitrariness of the choice how can we justify the choice of the truth value 0.24 over 0.23? I implausibility of the interpretation what does it mean for a sentence to be 1= true? Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 2 / 16
  • 3. Motivation (cont.) Unfolding the objection % the logic assigns a certain real number to sentences % how to measure exactly the value unique, exact real number as truth-value measurability in principle Diagnosis I point-wise valuation I numerical (cardinal) assignment Our proposal: ordinal foundation Pairwise valuations (‘being more or less true’) with no intensity Representation theorems Can infinite-valued valuations be proved to arise from truth-comparisons? Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 3 / 16
  • 4. Łukasiewicz infinite-valued logic I L = fp1; p2; : : : g I :, !, ? I SL I `Ł (Ł1) ! ( ! ) (Ł2) ( ! ) ! (( ! ) ! ( ! )) (Ł3) (: ! :) ! ( ! ) (Ł4) (( ! ) ! ) ! (( ! ) ! ) (MP) Standard truth-value semantics v : SL ! [0; 1] 1. v(?) = 0. 2. v(:) = 1 v() 3. v( ! ) = 1; if v() v(); 1 v() + v(); otherwise. Order-based semantics SL SL I ()def and : Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 4 / 16
  • 5. Structural constraints (A.1) is transitive (A.2) ? Soundness (A.3) ` =) Truth-condition for the implication (A.4) () ! Monotonicity constraints (A.5) 1 2; 1 2 =) 1 ! 1 2 ! 2 (A.6) =) : : Theorem If satisfies axioms (A.1)–(A.6) then there exists a Łukasiewicz valuation v : SL ! [0; 1] such that for all ; 2 SL: =) v() v(): Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 5 / 16
  • 6. Some consequences of the axioms 1. is reflexive 2. Define := : ! and
  • 7. := :(: :). If axiom (A.6) is satisfied, then axiom (A.5) is equivalent to the following: (A.50) 1 2; 1 2 =) 1
  • 8. 1 2
  • 9. 2 (A.500) 1 2; 1 2 =) 1 1 2 2 1. ` ! =) 2. ` $ =) 1. is an equivalence relation 2. 1 1; 2 2 =) 1 2 1 2 3. =) : : Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 6 / 16
  • 10. An algebra over (SL;) I SL== f [] j 2 SL g I [] = f j g ? := [?] := ? I : I [] := [:] I [] [] := [ ] I [] [] () 9i 2 []; i 2 [] i i I [] [] () [] ! [] = [] : and are well defined. The two orders, and , coincide. Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 7 / 16
  • 11. Relation to the Lindenbaum algebra ()def ` $ (SL;) (SL ;:;; 0;) (SL ; : ; ; ?;) q q f([]) = [] Lemma f(q()) = q(). f is well defined, onto and it is a homomorphism. Lemma (SL=; : ; ; ?) is a non-trivial MV-algebra. Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 8 / 16
  • 12. Theorem (Cignoli et al. 2000) If M is a non-trivial MV-algebra then there exists at least one homomorphism: m: M ! [0; 1]MV : q m (SL;) (SL=; : ; ; ?;) ([0; 1];:;; 0;) V I There exists V : SL ! [0; 1]. I V is a Łukasiewicz valuation. I V preserves , namely: =) V() V(): Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 9 / 16
  • 13. Corollaries (A.C) or (A.A) if and ? then 9n | {z } n Corollary If satisfies axioms (A.1)–(A.6) and (A.C) then there exists a unique valuation representing . Corollary If satisfies axioms (A.1)–(A.6), (A.C) and (A.A) then there exists a unique Łukasiewicz valuation v : SL ! [0; 1] such that for all ; 2 SL: () v() v(): Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 10 / 16
  • 14. Completeness I j= ()def for all satisfying (A.1)–(A.6) I j= ()def for all satisfying (A.1)–(A.6) if 8 2 then Soundness It easily follows from the axioms. Completeness 8 2 SL j= =) ` : I j= =) ` [Rose-Rosser, 1958] I j= =) j= Strong completeness 8 2 SL; 8 SL j= =) ` : Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 11 / 16
  • 15. Desirability of the axioms (A.1) is transitive (A.2) ? (A.3) ` =) (A.4) () ! (A.5) 1 2; 1 2 =) 1 ! 1 2 ! 2 (A.6) =) : : (A.C) or (A.A) if and ? then 9n | {z } n The axiom (A.3): ` =) i. abbreviation ii. soundness iii. non-justificative aim iv. generalization Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 12 / 16
  • 16. Generalization (future work) Why Łukasiewicz? The objection does not apply with the same strength to any logic with a [0; 1]-valued semantics Possible generalizations Remark Let v be a Łukasiewicz (Gödel, Product logic) valuation. The order defined by v ()def v() v() satisfies axioms (A.C), (A.1)–(A.6) where ` is the Łukasiewicz (Gödel, Product logic) deducibility relation. I sufficient conditions I a general case Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 13 / 16
  • 17. Conclusion Philosophical objection interpretation and measurability of ‘truth degrees’ Proposal ordinal foundation Main result If the relation SL2, intuitively interpreted as no more true than, satisfies specific conditions then there exists (at least) one infinite-valued valuation representing it Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 14 / 16
  • 18. Conclusion (cont.) Feedback If the alternatives can be compared ‘well enough’ then it is as if we attach a numerical valuation I the semantics based on the notion no more true than might be considered as an alternative semantics (adequate) which is compatible with the standard one I being an ordinal semantics, it is ‘immune’ to the artificial precision objection I axioms as desirable properties of the relation I plausibility and mathematical convenience Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 15 / 16
  • 19. References Roberto L. O. Cignoli, Italia M. L. D’Ottaviano and Daniele Mundici. Algebraic foundations of many-valued reasoning, Trends in Logic – Studia Logica Library, Kluwer Academic Publishers, 2000. Petr Hájek. Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, 1998. Petr Cintula, Petr Hájek, Carles Noguera (ed.) Handbook of Mathematical Fuzzy Logic - vol. 1 Studies in Logic, Mathematical Logic and Foundations, vol. 37, College Publications, London, 2011 Petr Cintula, Petr Hájek, Carles Noguera (ed.) Handbook of Mathematical Fuzzy Logic - vol. 2 Studies in Logic, Mathematical Logic and Foundations, vol. 38, College Publications, London, 2011 Rosanna Keefe. Theories of vagueness, Cambridge University Press, 2000. Rossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 16 / 16