rossellamarrano_vagueness

A qualitative perspective on vagueness and degrees oftruth

Rossella Marrano

Scuola Normale Superiore, Pisa

Based on a joint work with Hykel Hosni and Vincenzo Marra

Sydney, 18 December 2014

Rossella Marrano (SNS, Pisa) A qualitative perspective on vagueness and degrees of truth18/12/2014 1 / 31

Overview

The big picture

Logical and philosophical analysis of the notion of graded truth

I possible interpretations and possible models

I connection with graded belief

This talk

1. Vagueness and degrees of truth

2. Artificial precision objection

3. Qualitative perspective

4. A representation theorem for graded truth

5. Discussion and philosophical feedback


Vagueness and degrees of truth

Renoir, La vague, 1879

Vagueness

1. borderline cases2. no sharp boundaries

3. sorites paradoxes

Vagueness by degrees of truth

1. intermediate truth values2. graded membership

3. continuum of values

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 coded by(some) reals. [. . . ] We shall always take the set [0, 1] with its natural(standard) linear order. (Hájek, 1998)


Artificial precision objection

[Fuzzy logic] imposes artificial precision [. . . ]. [T]hough one is not obligedto require that a predicate either definitely applies or definitely does notapply, one is obliged to require that a predicate definitely applies to such-and-such, rather than to such-and-such other, degree (e.g. that a man 5 ft10 in tall belongs to tall to degree 0.6 rather than 0.5). (Haack, 1979)

One serious objection to [the many-valued approach] is that it really replacesvagueness with the most incredible and refined precision. (Tye, 1989)

[T]he degree theorist’s assignments impose precision in a form that is justas unacceptable as a classical true/false assignment. [. . . ] All predicationsof “is red” will receive a unique, exact value, but it seems inappropriate toassociate our vague predicate “red” with any particular exact function fromobjects to degrees of truth. For a start, what could determine which is thecorrect function, settling that my coat is red to degree 0.322 rather than0.321? (Keefe, 1998)


Unfolding the objection

% the logic assigns a certain real number to sentences

" unique, exact real number as truth-value

% how to measure exactly the value

" measurability in principle

Artificial precision objectionA semantics based on functions from sentences to degrees of truth coded byreal numbers misrepresents the phenomenon of vagueness

A possible answerSemantics based on pairwise evaluations


Qualitative perspective

I comparative judgments

I pairwise evaluation

I � ⊆ X2

Quantitative perspective

I numerical assignment

I point-wise evaluation

I Φ: X → R

The task is to isolate axioms that, on the one hand, are empirically and/orphilosophically acceptable for at least one important scientific interpreta-tion of the primitives and that, on the other hand, permit us to provemathematically that the structure is closely similar (usually, isomorphic orhomomorphic) to some numerical structure. (Luce & Narens, 1981)

General form of the representationNecessary and sufficient conditions on a relational structure 〈X,�〉 for theexistence of a(n equivalence class of a) real-valued function Φ such that for allx, y ∈ X

x � y ⇐⇒ Φ(x) ≤ Φ(y).


A qualitative perspective on degrees of truth

1. Quantitative truth: p is 0.7 trueI real-valued valuation functionsI Łukasiewicz clauses

2. Qualitative truth: p is more true than qI binary relation between sentencesI semantical notion (compare with Casari, 1987)

3. Representation results:can real-valued valuations be proved to arise from truth-comparisons?


Łukasiewicz real-valued logicLanguage

I L = {p, q, r, . . . }I ⊥I ¬, →I SL

Defined connectives

I > := ¬⊥I θ ⊕ φ := ¬θ → φ

I θ � φ := ¬(¬θ ⊕ ¬φ)

I . . .

Łukasiewicz logic

(Ł1) θ → (φ→ θ)

(Ł2) (θ → φ)→ ((φ→ χ)→ (θ → χ))

(Ł3) (¬θ → ¬φ)→ (φ→ θ)

(Ł4) ((θ → φ)→ φ)→ ((φ→ θ)→ θ)

(MP)

I `


Two evaluation methods

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 (‘no more true than’) � ⊆ SL × SLI θ ∼ φ⇐⇒def θ � φ and φ � θ.


Structural constraints

(A.1) � is reflexive and transitive

(A.2) ⊥ ≺ >

Soundness

(A.3) ` θ =⇒ θ ∼ >

Truth-condition for the implication

(A.4) θ → φ ∼ > =⇒ θ � φ

Monotonicity constraints

(A.5) θ � φ =⇒ ¬φ � ¬θ

(A.6) θ1 � θ2, φ1 � φ2 =⇒ θ1 → φ1 � θ2 → φ2

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

If axiom (A.5) is satisfied, then axiom (A.6) is equivalent to the following:

(A.6′) θ1 � θ2, φ1 � φ2 =⇒ θ1 ⊕ φ1 � θ2 ⊕ φ2

(A.6′′) θ1 � θ2, φ1 � φ2 =⇒ θ1 � φ1 � θ2 � φ2

1. θ � φ =⇒ θ → φ ∼ >

2. ` θ → φ =⇒ θ � φ

3. ` θ ↔ φ =⇒ θ ∼ φ

1. ∼ is an equivalence relation

2. θ ∼ φ =⇒ ¬φ ∼ ¬θ

3. θ1 ∼ φ1, θ2 ∼ φ2 =⇒ θ1 ⊕ θ2 ∼ φ1 ⊕ φ2


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.L) Either 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.L) then there exists a uniquevaluation representing �.

CorollaryIf � satisfies axioms (A.1)–(A.6), (A.L) 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).


Discussion and philosophical feedback

1. Soundness and completeness

2. Desirability of the axioms

3. Uniqueness of the representation

4. Artificial precision objection blocked?

5. Possible generalisations


Completeness

I Θ |=� φ⇐⇒def for all � satisfying (A.1)–(A.6) if ∀θ ∈ Θ θ ∼ > thenφ ∼ >

I |=� φ⇐⇒def for all � satisfying (A.1)–(A.6) φ ∼ >

SoundnessIt easily follows from the axioms.

Completeness

∀φ ∈ SL |=� φ =⇒ ` φ.

I |= φ =⇒ ` φ [Rose-Rosser, 1958]

I |=� φ =⇒ |= φ


Completeness

I Θ |=� φ⇐⇒def for all � satisfying (A.1)–(A.6) if ∀θ ∈ Θ θ ∼ > thenφ ∼ >

I |=� φ⇐⇒def for all � satisfying (A.1)–(A.6) φ ∼ >

SoundnessIt easily follows from the axioms.

Strong completeness

∀φ ∈ SL, ∀Θ ⊆ SL Θ |=� φ =⇒ Θ ` φ.

I Θ |=C φ =⇒ Θ ` φ [MV-chain completeness]

I Θ |=� φ =⇒ Θ |=C φ


Desirability of the axioms: structural constrains

Reflexivity θ � θTransitivity If θ � φ and φ � χ then θ � χNon-triviality ⊥ ≺ >Boundedness ⊥ � θ � >

Linearity Either θ � φ or φ � θ

I linearity is highly problematic:

I claim that again the connectedness axiom is incompatible with the natureof the vagueness of these comparisons: we cannot assume that there isalways a fact of the matter about which of two borderline sentences is moretrue. (Keefe, 2000)

I in the qualitative framework it can be questioned and droppedI there will be a family of valuations compatible with the order (the price

is non-constructivity)


Desirability of the axioms: logical constrains

(A.3) ` θ =⇒ θ ∼ >

I abbreviation (it can be replaced with a list of suitable conditions)I soundnessI non-justificative aimI generalisation to other (fuzzy) logics

(A.4) θ → φ ∼ > =⇒ θ � φ

I truth-condition for the implication both in classical and fuzzy logics:

In two-valued logic, the implication φ → ψ is true iff the truth-value of φis less than or equal to the truth-value of ψ. (Hájek, 1998)

I probabilistic analogue


Desirability of the axioms: monotonicity constrains

Monotonicity constrains reflect the truth-functionality assumption

(A.5) θ � φ =⇒ ¬φ � ¬θ

I negation is a non-increasing function

(A.6) θ1 � θ2, φ1 � φ2 =⇒ θ1 → φ1 � θ2 → φ2

I implication is non-increasing in the first component and non-decreasingin the second




(A.5) θ � φ =⇒ ¬φ � ¬θ


(A.6) θ1 � θ2, φ1 � φ2 =⇒ θ1 ⊕ φ1 � θ2 ⊕ φ2

I disjunction is non-decreasing in both components

I additivity




(A.5) θ � φ =⇒ ¬φ � ¬θ


(A.6) θ � φ =⇒ θ ⊕ χ � φ⊕ χ

I translation invariance


Desirability of the axioms: continuity constrain

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

� φ

I nothing infinitely less true

I it can be questioned and dropped

I weak representation result: there exists a valuation function whichalmost agrees with the order

N.B. non-idempotence of strong disjunction is crucial in this formulation


Uniqueness of the representation

Representation theoremIf � ⊆ X2 satisfies conditions Ax then there exists a function Φ: X → R suchthat for all x, y ∈ X

x � y ⇐⇒ Φ(x) ≤ Φ(y).

Depending on Ax, the function Φ will be unique up to a certain class oftransformations f : [0, 1]→ [0, 1] corresponding to the scales of measurementof the objects in X.

Ordinal scale monotone transformationsΦ(x) ≤ Φ(y) implies f(Φ(x)) ≤ f(Φ(y))

Interval scale linear transformationsf(Φ(x)) = αΦ(x) + b with α ∈ R++ and β ∈ R

Absolute scale identity transformation f(Φ(x)) = Φ(x)


Uniqueness of the representation

Absolute scaleThe Łukasiewicz valuation representing the order no more true than ascaptured by axioms (A.1)–(A.6), (A.L) and (A.A) is absolutely unique.

This holds irrespectively of the axioms:fixed endpoints rule out linear transformations

I v(⊥) = 0 implies β = 0 (fixed origin)

I v(>) = 1 implies α = 1 (fixed size of the unity)

truth-functionality of the negation rules out monotone transformations

I v(¬θ) = 1− v(θ) implies that f(0.5) = 0.5


Artificial precision objection blocked

Artificial precision objectionA semantics based on functions from sentences to degrees of truth coded byreal numbers misrepresents the phenomenon of vagueness

Main claimIn a qualitative framework this objection loses much of its force:

I pairwise evaluations do not require the same precision required innumerical assignments

I comparative judgments are less artificial or less arbitrary than absolutejudgments


Possible generalisations (future work)

I other (fuzzy) logics

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

θ �v φ⇐⇒def v(θ) ≤ v(φ)

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

I other consequence relations (es. logics preserving degrees)

I expected truth-value


Recap

ProposalQualitative perspective on vagueness and degrees of truth: semantics based onpairwise evaluations

Main result

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

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


Conclusion

The notion no more true than is defined axiomatically by means of desirableconditions.

The semantics based on the notion no more true than

I meets out intuitions on the connection between vagueness and a gradednotion of truth

I is an alternative semantics for the logic (sound and complete)

I yields the standard truth-value semantics (plausibility and mathematicalconvenience)

I unlike the standard oneI is strongly completeI can deal with the linearity objectionI is ‘immune’ to the artificial precision objection


Appendix

(i) Given a relation � define

Θ� := { θ ∈ SL | > � θ } .

(ii) Given a set Θ define

θ �Θ φ⇐⇒def Θ ` θ → φ.

Proposition

1. If � satisfies (A.1)–(A.6) then Θ� is deductively closed.

2. If Θ is deductively closed then �Θ satisfies axioms (A.1)–(A.6).


Λ �Λ Θ�Λ

(ii) (i)Λ = Θ�Λ

θ ∈ Θ�Λ ⇐⇒ > �Λ θ

⇐⇒ Λ ` > → θ

⇐⇒ Λ ` θ⇐⇒ θ ∈ Λ

v Θv �Θv

(i) (ii)v = �Θv

θ �Θv φ⇐⇒ Θv ` θ → φ

⇐⇒ θ → φ ∈ Θv

⇐⇒ > v θ → φ

⇐⇒ θ v φ


ReferencesEttore Casari.Comparative logics.Synthese 73(3): 421-449, 1987.

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.

Susan Haack.Do we need “fuzzy logic”?International Journal of Man-Machine Studies 11(4): 437–445, 1979.

Petr Hájek.Metamathematics of Fuzzy Logic,Kluwer Academic Publishers, 1998.

Rosanna Keefe.Theories of vagueness,Cambridge University Press, 2000.

R. Duncan Luce and Louis Narens.Axiomatic Measurement Theory.SIAM-AMS Proceedings, Volume 13, 1981.

Nicholas J. J. Smith.Vagueness and Degrees of TruthOxford University Press, 2008.

Michael Tye.Supervaluationism and the law of excluded middleAnalysis 49(3), 141–143, 1989.


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rossellamarrano_vagueness

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