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Vagueness, graded truth and pairwise valuations

Rossella Marrano

Scuola Normale Superiore, Pisa

The Future of Mathematical Fuzzy Logic Prague, 17 June 2016

MFL and vagueness

Rossella Marrano (SNS) The future of MFL 17/6/2016 1 / 15

MFL and vagueness

A possible diagnosis

The philosophical discussion on the role and significance of MFL remains focused on real-valued semantics and numerical degrees of truth.

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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. (Hájek, 1998)

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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. (Hájek, 1998)

The original cathedral Employing a well-understood and familiar mathematical structure in

• rder to
• 1. model the idea of truth as a graded notion,
• 2. provide a formal theory of vagueness.

Rossella Marrano (SNS) The future of MFL 17/6/2016 2 / 15

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. (Hájek, 1998)

The original cathedral Employing a well-understood and familiar mathematical structure in

• rder to
• 1. model the idea of truth as a graded notion,
• 2. provide a formal theory of vagueness.

The reality

• 1. DoT have been considered philosophically implausible,
• 2. MFL has been considered inadequate as a theory of vagueness.

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Intrinsic philosophical implausibility of DoT

First, it is necessary to say something about what a degree of truth is. Second, some account must be given of the source and justification

• f the numbers that are to be assigned as degrees. (Sainsbury, 1995)

Absence of some substantial philosophical account of what degrees

• f truth are. (Graff, 2000)

Questions about the nature of degrees of truth:

• 1. how can truth be measured to an exact extent?
• 2. how do we interpret the fact that a sentence is true to a certain

degree?

• 3. what are degrees of truth?

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Inadequacy of DoT

[T]he degree theorist’s assignments impose precision in a form that is just as unacceptable as a classical true/false assignment. [. . . ] All predications of “is red” will receive a unique, exact value, but it seems inappropriate to associate our vague predicate “red” with any particular exact function from objects to degrees of truth. For a start, what could determine which is the correct function, settling that my coat is red to degree 0.322 rather than 0.321? (Keefe, 1998)

Artificial precision objection

A semantics based on functions from sentences to degrees of truth coded by real numbers misrepresents the phenomenon of vagueness.

Rossella Marrano (SNS) The future of MFL 17/6/2016 4 / 15

Inadequacy of DoT

[T]he degree theorist’s assignments impose precision in a form that is just as unacceptable as a classical true/false assignment. [. . . ] All predications of “is red” will receive a unique, exact value, but it seems inappropriate to associate our vague predicate “red” with any particular exact function from objects to degrees of truth. For a start, what could determine which is the correct function, settling that my coat is red to degree 0.322 rather than 0.321? (Keefe, 1998)

Artificial precision objection

A semantics based on functions from sentences to degrees of truth coded by real numbers misrepresents the phenomenon of vagueness.

...and many other objections

Ambiguity of degrees objection, higher-order vagueness objection, linearity objection and truth-functionality objection.

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No easy way out

◮ Logic is concerned with valid inferences, rather than with concrete

truth-values assignments.

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No easy way out

◮ Logic is concerned with valid inferences, rather than with concrete

truth-values assignments.

◮ Structuralist position: all that matters are the structural

properties of the set.

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No easy way out

◮ Logic is concerned with valid inferences, rather than with concrete

truth-values assignments.

◮ Structuralist position: all that matters are the structural

properties of the set.

◮ Logic as modelling view: the unwanted precision of the

mathematical structure is treated as an artefact of the model.

Rossella Marrano (SNS) The future of MFL 17/6/2016 5 / 15

No easy way out

◮ Logic is concerned with valid inferences, rather than with concrete

truth-values assignments.

◮ Structuralist position: all that matters are the structural

properties of the set.

◮ Logic as modelling view: the unwanted precision of the

mathematical structure is treated as an artefact of the model. Until valuation functions remain the single most important element of the formal semantics, issues related to the philosophical status of truth values cannot be left out of consideration.

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Renovating the cathedral

Instead of concluding that

◮ graded truth is a meaningless notion, ◮ graded truth does not make a good theory of vagueness,

we could focus on the comparative aspect of graded truth.

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Renovating the cathedral

Instead of concluding that

◮ graded truth is a meaningless notion, ◮ graded truth does not make a good theory of vagueness,

we could focus on the comparative aspect of graded truth. Graded truth (preformal)

◮ quantitative: φ is true x, with x ∈ [0, 1], ◮ qualitative or comparative: φ is more true than ψ.

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Truth from comparison I

Shift in focus from pointwise valuations, assigning degrees of truth to sentences, to pairwise valuations based on comparative judgements of the form “the sentence φ is less (or more) true than the sentence ψ”.

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Truth from comparison I

Shift in focus from pointwise valuations, assigning degrees of truth to sentences, to pairwise valuations based on comparative judgements of the form “the sentence φ is less (or more) true than the sentence ψ”. Formalisations of graded truth

◮ pointwise valuations: v: SL → [0, 1], ◮ pairwise valuations: ⊆ SL2.

Rossella Marrano (SNS) The future of MFL 17/6/2016 7 / 15

Truth from comparison II

Main problem: representation results

Determine which conditions a pairwise valuation should satisfy in

• rder to guarantee the existence of a pointwise valuation v: SL → [0, 1]

representing it, namely such that for all φ, ψ ∈ SL φ ψ ⇒ v(φ) ≤ v(ψ).

Rossella Marrano (SNS) The future of MFL 17/6/2016 8 / 15

Truth from comparison II

Main problem: representation results

Determine which conditions a pairwise valuation should satisfy in

• rder to guarantee the existence of a pointwise valuation v: SL → [0, 1]

representing it, namely such that for all φ, ψ ∈ SL φ ψ ⇒ v(φ) ≤ v(ψ). Pairwise valuations act as (i) method for evaluating sentences of a given formal language, (ii) foundation for the standard truth-value approach. If sentences can be compared ‘well enough’ with respect to their truth then it is as if we attach them a specific truth value.

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Benefits

◮ Catching up with the mathematical work in MFL that emphasises

the comparative aspect of fuzzy logics: real-valued algebras are not the only intended semantics for these logics!

◮ Philosophical value:

◮ no recourse to problematic objects, ◮ structuralism taken seriously, ◮ more fundamental level.

◮ Instrumental value: representation results reintroduce DoT and

make the whole machinery of MFL available.

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Degrees of truth explained away

Philosophical underpinning

The numerical aspect of mapping sentences into degrees of truth can be considered secondary with respect to the comparative aspect, which consists in ranking sentences by performing pairwise comparisons.

Rossella Marrano (SNS) The future of MFL 17/6/2016 10 / 15

Degrees of truth explained away

Philosophical underpinning

The numerical aspect of mapping sentences into degrees of truth can be considered secondary with respect to the comparative aspect, which consists in ranking sentences by performing pairwise comparisons. Feedback on the nature of degrees of truth:

◮ they are formal constructions encoding the relative positions of

sentences with respect to the primitive relation more or less true than;

◮ they are interpreted as possible measures (or cardinalisations) of a

comparative notion of truth.

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Vagueness, graded truth and pairwise valuations

There is nothing essentially numerical about graded truth, nothing that forces us to formalise it by using real numbers.

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Vagueness, graded truth and pairwise valuations

There is nothing essentially numerical about graded truth, nothing that forces us to formalise it by using real numbers. The semantics based on pairwise valuations

◮ meets our intuitions on the connection between vagueness and

graded truth,

◮ is an alternative semantics for the logic (strongly sound and

complete),

◮ mitigates some of the weak points of the degree-theoretic

semantics,

◮ provides a possible formal theory of vagueness.

Rossella Marrano (SNS) The future of MFL 17/6/2016 11 / 15

Sorites and pairwise judgements

Shift of focus from the predicates P and non-P to the binary relation ‘more or less P than’ defined over the objects in the domain of P.

[P]roponents of the standard analysis conceive of borderline cases in terms of a certain kind of ordering. They suppose that for any pre- dicate ‘Φ’ having borderline cases, there is some linear ordering of items (values) on a dimension decisive of the application of ‘Φ’, pro- gressing from an item that is definitely Φ to an item that is definitely not-Φ. Call such an ordering a Φ-ordering. (Raffman, 2014)

If P is vague, then the P-ordering is non-trivial, linear, with some intermediate elements (borderline cases) and there may be indefinitely many steps between the top and the bottom element. Sorites as a chain in which each object is no P-er than the previous

• ne, with at least some significant differences.

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Adjacency

Adjacency: For any objects x and y, if they are adjacent in the P-ordering, then ‘P(x)’ and ‘P(y)’ are adjacent in the truth ordering or they are in the same equivalence class, ‘P(x)′ ∼ ‘P(y)’.

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Adjacency = Closeness (Smith, 2005)

Adjacency: For any objects x and y, if they are adjacent in the P-ordering, then ‘P(x)’ and ‘P(y)’ are adjacent in the truth ordering or they are in the same equivalence class, ‘P(x)′ ∼ ‘P(y)’. Closeness: For any objects x and y, if they are very close in P-relevant respects, then ‘P(x)’ and ‘P(y)’ are very close in respect of truth.

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Adjacency = Closeness (Smith, 2005)

Adjacency: For any objects x and y, if they are adjacent in the P-ordering, then ‘P(x)’ and ‘P(y)’ are adjacent in the truth ordering or they are in the same equivalence class, ‘P(x)′ ∼ ‘P(y)’. Closeness: For any objects x and y, if they are very close in P-relevant respects, then ‘P(x)’ and ‘P(y)’ are very close in respect of truth. We are not justified to conclude in general for any given predicate P that if two elements xi and xj are adjacent with respect to the P-ordering then they are close in P-relevant respects, unless we know that P is vague.

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Not a definition of vagueness

◮ No notion of almost true, closeness, small step can be expressed in

comparative terms,

◮ the topology over real numbers is necessary in order to be able to

formulate distance considerations. The theory does not have the resources to distinguish fine-graded series from coarse-graded series and to distinguish vague predicates from graded predicates that are not vague (not even from precise predicates!) Pairwise valuations provide a theory of graded predicates, and if we consider those that are vague, then it provides a formal theory of vagueness.

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What’s next?

◮ Better understanding of graded truth as philosophical notion:

◮ is it subjective, objective, intersubjective? ◮ how is it related to graded belief?

◮ Fruitful generalisations:

◮ predicative case, ◮ graded consequence relations.

◮ Build bridges with psychology and improve on the cognitive

plausibility of the model.

Rossella Marrano (SNS) The future of MFL 17/6/2016 15 / 15

What’s next?

◮ Better understanding of graded truth as philosophical notion:

◮ is it subjective, objective, intersubjective? ◮ how is it related to graded belief?

◮ Fruitful generalisations:

◮ predicative case, ◮ graded consequence relations.

◮ Build bridges with psychology and improve on the cognitive

plausibility of the model.

Děkuji!

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