rossellamarrano_latd2014

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 1as maximum and 0 as minimum. Thus truth degrees will be codedby (some) reals. [. . . ] We shall always take the set [0, 1] with itsnatural (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/π true?


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 foundationPairwise valuations (‘being more or less true’) with no intensity

Representation theorems Can infinite-valued valuations be proved to arisefrom truth-comparisons?


Łukasiewicz infinite-valued logicI L = {p1, p2, . . . }I ¬, →, ⊥I SL

I `Ł

(Ł1) θ → (φ→ θ)

(Ł2) (θ → φ)→ ((φ→ χ)→ (θ → χ))

(Ł3) (¬θ → ¬φ)→ (φ→ θ)

(Ł4) ((θ → φ)→ φ)→ ((φ→ θ)→ θ)

(MP)

Standard truth-value semantics v : SL → [0, 1]

1. v(⊥) = 0.

2. v(¬θ) = 1− v(θ)

3. v(θ → φ) =

{1, if v(θ) ≤ v(φ);1− v(θ) + v(φ), otherwise.

Order-based semantics � ⊆ SL × SLI θ ∼ φ⇐⇒def θ � φ and φ � θ.


Structural constraints

(A.1) � is transitive

(A.2) ⊥ ≺ >

Soundness

(A.3) ` θ =⇒ θ ∼ >

Truth-condition for the implication

(A.4) θ � φ⇐⇒ θ → φ ∼ >

Monotonicity constraints

(A.5) θ1 � θ2, φ1 � φ2 =⇒ θ1 → φ1 � θ2 → φ2

(A.6) θ � φ =⇒ ¬φ � ¬θ

TheoremIf � satisfies axioms (A.1)–(A.6) then there exists a Łukasiewicz valuationv : SL → [0, 1] such that for all θ, φ ∈ SL:

θ � φ =⇒ v(θ) ≤ v(φ).


Some consequences of the axioms

1. � is reflexive

2. θ � >

Define θ ⊕ φ := ¬θ → φ and θ � φ := ¬(¬θ ⊕ ¬φ). If axiom (A.6) is satisfied,then axiom (A.5) is equivalent to the following:

(A.5′) θ1 � θ2, φ1 � φ2 =⇒ θ1 � φ1 � θ2 � φ2(A.5′′) θ1 � θ2, φ1 � φ2 =⇒ θ1 ⊕ φ1 � θ2 ⊕ φ2

1. ` θ → φ =⇒ θ � φ

2. ` θ ↔ φ =⇒ θ ∼ φ

1. ∼ is an equivalence relation

2. θ1 ∼ φ1, θ2 ∼ φ2 =⇒ θ1 ⊕ θ2 ∼ φ1 ⊕ φ23. θ ∼ φ =⇒ ¬φ ∼ ¬θRossella Marrano (SNS) Ordinal foundation for Łukasiewicz semantics 17/07/2014 6 / 16

An algebra over (SL,�)

I SL/∼= { [θ]∼ | θ ∈ SL }I [θ]∼ = { φ | φ ∼ θ }

I∼⊥ := [⊥]∼ := ⊥

I∼¬[θ]∼ := [¬θ]∼

I [θ]∼∼⊕ [φ]∼ := [θ ⊕ φ]∼

I [θ]∼ ≤∼ [φ]∼ ⇐⇒ ∃θi ∈ [θ]∼, φi ∈ [φ]∼ θi � φiI [θ]∼ ≤ [φ]∼ ⇐⇒ [θ]∼

∼→ [φ]∼ = [>]∼

∼¬ and∼⊕ are well defined.

The two orders, ≤∼ and ≤, coincide.


Relation to the Lindenbaum algebra

θ ≡ φ⇐⇒def ` θ ↔ φ

(SL,�) (SL≡ ,¬,⊕, 0,≤)

(SL∼ ,∼¬,∼⊕,∼⊥,≤∼)

q≡

q∼

f([θ]≡) = [θ]∼

Lemmaf(q≡(θ)) = q∼(θ). f is well defined, onto and it is a homomorphism.

Lemma

(SL/∼,∼¬,∼⊕,∼⊥) is a non-trivial MV-algebra.


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 .

(SL,�) (SL/∼,∼¬,∼⊕,∼⊥,≤∼) ([0, 1],¬,⊕, 0,≤)

q∼ m

V�

I There exists V� : SL → [0, 1].

I V� is a Łukasiewicz valuation.

I V� preserves �, namely: θ � φ =⇒ V�(θ) ≤ V�(φ).


Corollaries

(A.C) θ � φ or φ � θ

(A.A) if θ � φ and θ � ⊥ then ∃n θ ⊕ · · · ⊕ θ︸︷︷︸n

� φ

CorollaryIf � satisfies axioms (A.1)–(A.6) and (A.C) then there exists a uniquevaluation representing �.

CorollaryIf � satisfies axioms (A.1)–(A.6), (A.C) and (A.A) then there exists a uniqueŁukasiewicz valuation v : SL → [0, 1] such that for all θ, φ ∈ SL:

θ � φ⇐⇒ v(θ) ≤ v(φ).


Desirability of the axioms

(A.1) � is transitive

(A.2) ⊥ ≺ >(A.3) ` θ =⇒ θ ∼ >(A.4) θ � φ⇐⇒ θ → φ ∼ >(A.5) θ1 � θ2, φ1 � φ2 =⇒ θ1 → φ1 � θ2 → φ2

(A.6) θ � φ =⇒ ¬φ � ¬θ

(A.C) θ � φ or φ � θ(A.A) if θ � φ and θ � ⊥ then ∃n θ ⊕ · · · ⊕ θ︸︷︷︸

n

� φ

The axiom (A.3): ` θ =⇒ θ ∼ >

i. abbreviation

ii. soundness

iii. non-justificative aim

iv. generalizationRossella 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

RemarkLet v be a Łukasiewicz (Gödel, Product logic) valuation. The order defined by

θ �v φ⇐⇒def v(θ) ≤ v(φ)

satisfies axioms (A.C), (A.1)–(A.6) where ` is the Łukasiewicz (Gödel,Product logic) deducibility relation.

I sufficient conditions

I a general case


Conclusion

Philosophical objectioninterpretation and measurability of ‘truth degrees’

Proposalordinal foundation

Main result

If the relation � ⊆ SL2, intuitively interpreted as no more true than, satisfiesspecific conditions then there exists (at least) one infinite-valued valuationrepresenting it


Conclusion (cont.)

Feedback

If the alternatives can be compared ‘well enough’ then it is as if we attach anumerical valuation

I the semantics based on the notion no more true than might beconsidered as an alternative semantics (adequate) which is compatiblewith the standard one

I being an ordinal semantics, it is ‘immune’ to the artificial precisionobjection

I axioms as desirable properties of the relation

I plausibility and mathematical convenience


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. 1Studies 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. 2Studies 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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