Burning Issue presentation of Zhazgul N. , Cycle 54
RossellaMarrano_LATD2014
1. Ordinal foundation for Łukasiewicz semantics
Rossella Marrano
Scuola Normale Superiore, Pisa
LATD 2014, Vienna, 17 July 2014
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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?
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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?
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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 :
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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():
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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.
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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.
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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():
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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():
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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= =) ` :
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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
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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
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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
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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
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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.
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