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. Rossella Marrano (SNS) The future of MFL 17/6/2016 1 / 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) 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 order 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 order 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. Rossella Marrano (SNS) The future of MFL 17/6/2016 2 / 15
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 of the numbers that are to be assigned as degrees. (Sainsbury, 1995) Absence of some substantial philosophical account of what degrees of 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? Rossella Marrano (SNS) The future of MFL 17/6/2016 3 / 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. 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. Rossella Marrano (SNS) The future of MFL 17/6/2016 4 / 15
No easy way out ◮ Logic is concerned with valid inferences, rather than with concrete truth-values assignments. 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. 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. 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. Rossella Marrano (SNS) The future of MFL 17/6/2016 5 / 15
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. Rossella Marrano (SNS) The future of MFL 17/6/2016 6 / 15
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 . Rossella Marrano (SNS) The future of MFL 17/6/2016 6 / 15
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 ”. Rossella Marrano (SNS) The future of MFL 17/6/2016 7 / 15
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: � ⊆ SL 2 . 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 order 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 order 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. Rossella Marrano (SNS) The future of MFL 17/6/2016 8 / 15
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. Rossella Marrano (SNS) The future of MFL 17/6/2016 9 / 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. 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. Rossella Marrano (SNS) The future of MFL 17/6/2016 10 / 15
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. Rossella Marrano (SNS) The future of MFL 17/6/2016 11 / 15
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