Handling vagueness in logic, via algebras and games. Lecture 1.

Handling vagueness in logic, via algebras and games. Lecture 1. Serafina Lapenta and Diego Valota S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 1/43

Handbook of Mathematical Fuzzy Logic. Volume 1-2-3. Studies in Logic, Mathematical Logic and Foundations. College Publications, London. P. Cintula, P. Hájek, and C. Noguera. Volume 1-2. Numbers 37-38, 2011. P. Cintula, C. G. Fermüller, and C. Noguera. Volume 3. Number 58, 2016. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 2/43

Lecture 1. From precise to vague predicates. Definition and theories of vagueness. Degrees of truth. N.J.J. Smith. Fuzzy logics in theories of vagueness. In Handbook of Math. Fuzzy Logic. Vol. 3, pages 1237 1281. Lecture 2. Introduction to mathematical fuzzy logic. The logics BL and MTL and related algebraic structures. L. Běhounek, P. Cintula and C. Noguera. Introduction to mathematical fuzzy logic. In Handbook of Math. Fuzzy Logic. Vol. 1, pages 1 101. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 3/43

Lecture 3. Łukasiewicz logic: motivations behind it and definitions. Vague predicates in Łukasiewicz logic, MV-algebras and Łukasiewicz-Moisil algebras. A. Di Nola and I. Leuştean. Łukasiewicz Logic and MV-Algebras. In Handbook of Math. Fuzzy Logic. Vol. 2, pages 469 584. V. Boicescu, A. Filipoiu, G. Georgescu, and S. Rudeanu. Łukasiewicz-Moisil Algebras. Annals of discrete mathematics. North-Holland, 1991. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 4/43

Lecture 4. Hintikka s Games for classical logics. Giles s Games for many-valued logics. C. G. Fermüller. Dialogue semantic games for fuzzy logics. In P. Cintula, C. G. Fermüller, and C. Noguera, editors, Handbook of Math. Fuzzy Logic. Vol. 3, pages 969 1028. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 5/43

Lecture 5. Ulam-Rényi s Games for Łukasiewicz logic. Akinator for Łukasiewicz-Moisil logic. F. Cicalese and F. Montagna. Ulam-rényi game based semantics for fuzzy logics. In Handbook of Math. Fuzzy Logic. Vol. 3, pages 1029 1062. D. Diaconescu and I. Leuştean. Towards game semantics for nuanced logics. In FUZZ-IEEE 2017, IEEE Int. Conference on Fuzzy Systems, https://doi.org/10.1109/fuzz-ieee.2017.8015600, 2017. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 6/43

Serafina Lapenta Department of Mathematics, Università degli Studi di Salerno, Via Giovanni Paolo II 132, 84084, Fisciano, Italy email : slapenta@unisa.it homepage: http://serafinalapenta.weebly.com/ Diego Valota Department of Computer Science, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milano, Italy email : valota@di.unimi.it homepage: http://homes.di.unimi.it/~valota/ S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 7/43

From crisp to fuzzy P. Cintula and C. Noguera, A gentle introduction to MFL. http://www2.cs.cas.cz/~cintula/mfl.html S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 8/43

From crisp to fuzzy P. Cintula and C. Noguera, A gentle introduction to MFL. http://www2.cs.cas.cz/~cintula/mfl.html For the contest of this course, logic is the science that studies correct reasoning What is then a correct reasoning? Consider the following If God exists, He must be good and omnipotent. If God was good and omnipotent, He would not allow human suffering. But, there is human suffering. Therefore, God does not exist. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 8/43

From crisp to fuzzy P. Cintula and C. Noguera, A gentle introduction to MFL. http://www2.cs.cas.cz/~cintula/mfl.html For the contest of this course, logic is the science that studies correct reasoning What is then a correct reasoning? Consider the following If God exists, He must be good and omnipotent. If God was good and omnipotent, He would not allow human suffering. But, there is human suffering. Therefore, God does not exist. Is this correct reasoning? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 8/43

Let s frame this argument in a formal system! Denumerable set of symbols: Var = {p, q, r, s,... p 1, p 2,... } called propositional variables. The language of CPC is given by (negation), (disjunction), (conjunction), (implication), (, ) (parenthesis) every symbol in Var. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 9/43

Classical propositional logic: syntax We look at Form, inductively defined by any element of Var belongs to Form, if ϕ Form, then ( ϕ) Form, if ϕ, ψ Form, then (ϕ ψ) Form, (ϕ ψ) Form, (ϕ ψ) Form. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 10/43

Classical propositional logic: semantics Bi-valence Principle Every proposition is either true or false. Assigning a truth value A 2-evaluation is a mapping e from Form to {0, 1} such that 1. e( ϕ) = 1 e(ϕ), 2. e(ϕ ψ) = max(e(ϕ), e(ψ)), 3. e(ϕ ψ) = min(e(ϕ), e(ψ)), 1 e(ϕ) e(ψ) 4. e(ϕ ψ) =. 0 otherwise S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 11/43

From crisp to fuzzy Let us fix classical logic as logic of choice, one can argue that correct reasoning has to be logical consequence S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 12/43

From crisp to fuzzy Let us fix classical logic as logic of choice, one can argue that correct reasoning has to be logical consequence Given Γ Form and ϕ Form, the deduction of ϕ from Γ is a correct reasoning if Γ = ϕ. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 12/43

From crisp to fuzzy Let us fix classical logic as logic of choice, one can argue that correct reasoning has to be logical consequence Given Γ Form and ϕ Form, the deduction of ϕ from Γ is a correct reasoning if Γ = ϕ. We are saying that there is no interpretation for which the premises are true and the the conclusion false. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 12/43

Some examples Modus ponens It is correct reasoning in CPC! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 13/43

Some examples Modus ponens It is correct reasoning in CPC! Abduction Abduction allows to deduce the premises from the conclusion, i.e. p q q p. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 13/43

Some examples Modus ponens It is correct reasoning in CPC! Abduction Abduction allows to deduce the premises from the conclusion, i.e. p q q p. It is not correct reasoning: take e(p) = 0 and e(q) = 1. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 13/43

So, does God exists? A formalization p= God exists, q= God is good, r= God is omnipotent, s= There is human suffering. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 14/43

So, does God exists? A formalization p= God exists, q= God is good, r= God is omnipotent, s= There is human suffering. Thus, the we get p q r q r s s p. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 14/43

p q r q r s s p. If e(p q r) = 1, e(q r s) = 1 and e(s) = 1, then necessarily e(p) = 0. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 15/43

p q r q r s s p. If e(p q r) = 1, e(q r s) = 1 and e(s) = 1, then necessarily e(p) = 0. Thus it is correct reasoning, but is it really a proof that god does not exist? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 15/43

p q r q r s s p. If e(p q r) = 1, e(q r s) = 1 and e(s) = 1, then necessarily e(p) = 0. Thus it is correct reasoning, but is it really a proof that god does not exist? Nope! We only proved that from true premises we get true conclusions. Where is the catch? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 15/43

When {0, 1} is not enough If you cut one head off of a two headed man, have you decapitated him? What is the maximum height of a short man? When does a fertilized egg develop into a person? Dictionary of Philosophy and Psychology s entry for vague (1902): A proposition is vague when there are possible states of things concerning which it is intrinsically uncertain whether, had they been contemplated by the speaker, he would have regarded them as excluded or allowed by the proposition. By intrinsically uncertain we mean not uncertain in consequence of any ignorance of the interpreter, but because the speaker s habits of language were indeterminate. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 16/43

From crisp to fuzzy Due to the bivalence principle, we can easily draw a line between those object to which a predicate applies and all the others. We call such predicates crisps. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 17/43

From crisp to fuzzy Due to the bivalence principle, we can easily draw a line between those object to which a predicate applies and all the others. We call such predicates crisps. Examples are prime numbers, monotonic functions, divisible groups... basically, most mathematical objects are ok for 0/1 predicates. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 17/43

From crisp to fuzzy Due to the bivalence principle, we can easily draw a line between those object to which a predicate applies and all the others. We call such predicates crisps. Examples are prime numbers, monotonic functions, divisible groups... basically, most mathematical objects are ok for 0/1 predicates. It can be argued that classical logic is better suited to capture correct reasoning in mathematics... thus what about the "real world"? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 17/43

From crisp to fuzzy Vague predicated vs precise predicates What is a vague predicate? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 18/43

From crisp to fuzzy Vague predicated vs precise predicates What is a vague predicate? Good question! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 18/43

From crisp to fuzzy Vague predicated vs precise predicates What is a vague predicate? Good question! Think of (monadic) predicated such as "Tall", "Red", "Rich", "Poor"... predicates from natural language Intuitively, you would agree with me in saying that they are all vague. But why? What they have in common? How can we formalize the notion of a vague predicate? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 18/43

From crisp to fuzzy Take a vague predicate V. Which are the defining features? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 19/43

From crisp to fuzzy Take a vague predicate V. Which are the defining features? 1. V admits borderline cases over the intended domain of interpretation: there is a c in the domain of the interpretation such that is not clear if V (c) holds or V (c) holds. Example: You have a group of people and you want to separate the Tall ones from the non-tall. You will have clear cases of Tallness, clear cases of non-tallness, but you will also have cases you cannot place in in Tall nor in non-tall. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 19/43

From crisp to fuzzy Take a vague predicate V. Which are the defining features? 1. V admits borderline cases over the intended domain of interpretation: there is a c in the domain of the interpretation such that is not clear if V (c) holds or V (c) holds. Example: You have a group of people and you want to separate the Tall ones from the non-tall. You will have clear cases of Tallness, clear cases of non-tallness, but you will also have cases you cannot place in in Tall nor in non-tall. When asked whether a person is tall, we tend to react with some sort of hedging response: sort of, a shrug, etc. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 19/43

From crisp to fuzzy 2. V lacks sharp boundaries over the intended domain of interpretation, i.e. there is no clearly defined boundary separating the extension of V ( ) from the one of its anti-extension. Example: if I put a pin on a map, I can t draw a circle of all the places that are near my pin. I can undoubtedly do it for all the places that are within 1km from the pin. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 20/43

From crisp to fuzzy 3. V is susceptible to a Sorites series over the intended domain of interpretation. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 21/43

From crisp to fuzzy 3. V is susceptible to a Sorites series over the intended domain of interpretation. There is sequence of c 1,..., c n in the domain such that it is clear that V (c 1 ) holds, it is clear that V (c n ) does not hold and it is at least plausible that V (c i ) V (c i+1 ) holds. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 21/43

From crisp to fuzzy 3. V is susceptible to a Sorites series over the intended domain of interpretation. There is sequence of c 1,..., c n in the domain such that it is clear that V (c 1 ) holds, it is clear that V (c n ) does not hold and it is at least plausible that V (c i ) V (c i+1 ) holds. Example: A man who has no money is poor. If a poor man earns one euro, he remains poor. Therefore, a man who has one million euros is poor. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 21/43

From crisp to fuzzy A man who has no money is poor. If a poor man earns one euro, he remains poor. Therefore, a man who has one million euros is poor. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 22/43

From crisp to fuzzy A man who has no money is poor. If a poor man earns one euro, he remains poor. Therefore, a man who has one million euros is poor. P(0) is clearly true, P(1000000) is clearly false, each P(n) P(n + 1) is correct reasoning: we use MP! Where is the issue? The truth fades little by little from P(0) top(1000000) Classical logic can t see it because of the bivalence principle! There is no clear jump from a true P(k) to a false P(k + 1). S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 22/43

A remark We have characterized a vague predicate by these 3 distinctive features: they give rise to borderline cases, their extensions have blurry boundaries, and they generate Sorite s paradoxes. These three features are closely related to each other, but they are not merely three ways of saying essentially the same thing! One can say that fuzzy boundaries are caused by the possibility to have borderline cases and viceversa! It can be argued that the central issue is that borderline cases are not sharply bounded S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 23/43

When vagueness is not vagueness In natural language, vagueness is associated with other phenomena: S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 24/43

When vagueness is not vagueness In natural language, vagueness is associated with other phenomena: Uncertainty S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 24/43

When vagueness is not vagueness In natural language, vagueness is associated with other phenomena: Uncertainty it is an epistemic phenomenon: the speaker does not know if x had the property P. E.g. "P = bearfast" applies to all xs that move faster than any polar bear moved on 11th January 1904. There are clear cases of P (an airplane) and clear cases of non-p (a car that is parked), but there are borderline cases! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 24/43

When vagueness is not vagueness In natural language, vagueness is associated with other phenomena: Uncertainty it is an epistemic phenomenon: the speaker does not know if x had the property P. E.g. "P = bearfast" applies to all xs that move faster than any polar bear moved on 11th January 1904. There are clear cases of P (an airplane) and clear cases of non-p (a car that is parked), but there are borderline cases! P is not a vague predicate S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 24/43

When vagueness is not vagueness In natural language, vagueness is associated with other phenomena: Uncertainty it is an epistemic phenomenon: the speaker does not know if x had the property P. E.g. "P = bearfast" applies to all xs that move faster than any polar bear moved on 11th January 1904. There are clear cases of P (an airplane) and clear cases of non-p (a car that is parked), but there are borderline cases! P is not a vague predicate indeed, it does not generate a Sorite series. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 24/43

When vagueness is not vagueness Ambiguity S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 25/43

When vagueness is not vagueness Ambiguity when a predicate is subject to different interpretations. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 25/43

When vagueness is not vagueness Ambiguity when a predicate is subject to different interpretations. E.g. heavy is either "of great weight" or "serious, important". So "Mark is an heavy person" can be ambiguous. What s the distinction with vagueness? Vagueness is still present after the disambiguation on "heavy". Context sensitivity when the extension of a P depends on the context. E.g., "P = Tall" has different extensions among tennis players and basketball players S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 25/43

When vagueness is not vagueness Ambiguity when a predicate is subject to different interpretations. E.g. heavy is either "of great weight" or "serious, important". So "Mark is an heavy person" can be ambiguous. What s the distinction with vagueness? Vagueness is still present after the disambiguation on "heavy". Context sensitivity when the extension of a P depends on the context. E.g., "P = Tall" has different extensions among tennis players and basketball players we restrict P to a comparison class. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 25/43

When vagueness is not vagueness Ambiguity when a predicate is subject to different interpretations. E.g. heavy is either "of great weight" or "serious, important". So "Mark is an heavy person" can be ambiguous. What s the distinction with vagueness? Vagueness is still present after the disambiguation on "heavy". Context sensitivity when the extension of a P depends on the context. E.g., "P = Tall" has different extensions among tennis players and basketball players we restrict P to a comparison class. Many vague predicates are context sensitive, but context sensitivity and vagueness are different phenomena: vagueness arises even on a single occasion of use. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 25/43

When vagueness is not vagueness Generality S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 26/43

When vagueness is not vagueness Generality by way of example, when we reply to a question with a vague answer. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 26/43

When vagueness is not vagueness Generality by way of example, when we reply to a question with a vague answer. E.g. replying "I was born in the last century" instead of "I was born in 1987". It is different from vagueness as it does not generate a Sorite series: there is not a clear truth of P(t) P(t + 1), if we consider people that are born one minute apart on 31st December 1999. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 26/43

Theories of vagueness What do we want? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 27/43

Theories of vagueness What do we want? For our purpose, any theory of vagueness should answer to what is the meaning of a vague predicate? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 27/43

Theories of vagueness What do we want? For our purpose, any theory of vagueness should answer to what is the meaning of a vague predicate? semantics S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 27/43

Theories of vagueness What do we want? For our purpose, any theory of vagueness should answer to what is the meaning of a vague predicate? semantics how should we reason in presence of vagueness? syntax A theory of vagueness should solve a Sorite paradox! locate the error in the Sorite argument explain why it is a paradox rather than a mistake... why a competent speaker finds the argument compelling but not convincing? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 27/43

Non-fuzzy theories of vagueness Epistemicist solution: Vagueness is a problem of ignorance. All predicates are crisp, but we are unable to know the exact extension of a vague predicate. Some implication P(k) P(k + 1) is false. A remark: bearfast in the example of uncertainty does not clash with the epistemic approach! It gives an example of a whole class of objects for which no speaker can know if it belongs to the extension of P. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 28/43

Supervaluationist solution: The meaning of vague predicate is the set of its precisifications (possible ways to make it crisp). Truth is supertruth, i.e. true under all precisifications. Some implication P(k) P(k + 1) is false. Vague predicates do not have a univocal meaning. A vague language is a set of crisp languages. For every utterance of a sentence involving a vague predicate, pragmatical conventions endow it with some particular crisp meaning. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 29/43

Supervaluationist solution: The meaning of vague predicate is the set of its precisifications (possible ways to make it crisp). Truth is supertruth, i.e. true under all precisifications. Some implication P(k) P(k + 1) is false. Vague predicates do not have a univocal meaning. A vague language is a set of crisp languages. For every utterance of a sentence involving a vague predicate, pragmatical conventions endow it with some particular crisp meaning. Some implication P(k) P(k + 1) is false. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 29/43

Supervaluationist solution: The meaning of vague predicate is the set of its precisifications (possible ways to make it crisp). Truth is supertruth, i.e. true under all precisifications. Some implication P(k) P(k + 1) is false. Vague predicates do not have a univocal meaning. A vague language is a set of crisp languages. For every utterance of a sentence involving a vague predicate, pragmatical conventions endow it with some particular crisp meaning. Some implication P(k) P(k + 1) is false. This is closely related to context sensitivity, but it is not the same! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 29/43

Epistemicist solution Vagueness is a problem of ignorance. Some implication P(k) P(k + 1) is false. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 30/43

Epistemicist solution Vagueness is a problem of ignorance. Some implication P(k) P(k + 1) is false. We don t know for which k the implication is false, and we assume that there all implications are true. This implies a split between the meaning of the predicate and the actual usage of the same predicate by a competent speaker! Usage should determine meaning. meaning true false use assert hedge deny S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 30/43

Epistemicist solution Pros: Classical reasoning is correct even in presence of vague predicates. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 31/43

Epistemicist solution Pros: Classical reasoning is correct even in presence of vague predicates. Cons: Location Problem: Why P(k) P(k + 1) is false for that particular k? Meaning-Usage: meaning is m-partite, usage is n-partite. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 31/43

Supervaluationist solution The meaning of vague predicate is the set of its precisifications. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 32/43

Supervaluationist solution The meaning of vague predicate is the set of its precisifications. One option is to precisify via classical sharpening. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 32/43

Supervaluationist solution The meaning of vague predicate is the set of its precisifications. One option is to precisify via classical sharpening. Start with a 3-valued evaluation: V 3 : Var {0,, 1}. Taken a classical V 2 : Var {0, 1}, we say that V 2 extends V 3 iff V 3 (P) = 0 (V 3 (P) = 1) then V 2 (P) = 0 (V 2 (P) = 1). Acceptable extensions allow to precisify all vague predicates at once. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 32/43

Supervaluationist solution The meaning of vague predicate is the set of its precisifications. One option is to precisify via classical sharpening. Start with a 3-valued evaluation: V 3 : Var {0,, 1}. Taken a classical V 2 : Var {0, 1}, we say that V 2 extends V 3 iff V 3 (P) = 0 (V 3 (P) = 1) then V 2 (P) = 0 (V 2 (P) = 1). Acceptable extensions allow to precisify all vague predicates at once. Then V 3 (P) = 1 iff V 2 (P) = 1 for any possible extension. The same for V 3 (P) = 0. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 32/43

Supervaluationist solution The meaning of vague predicate is the set of its precisifications. One option is to precisify via classical sharpening. Start with a 3-valued evaluation: V 3 : Var {0,, 1}. Taken a classical V 2 : Var {0, 1}, we say that V 2 extends V 3 iff V 3 (P) = 0 (V 3 (P) = 1) then V 2 (P) = 0 (V 2 (P) = 1). Acceptable extensions allow to precisify all vague predicates at once. Then V 3 (P) = 1 iff V 2 (P) = 1 for any possible extension. The same for V 3 (P) = 0. If there exist extensions V 2, W 2 such that V 2 (P) = 1 and W 2 (P) = 0, then V 3 (P) =. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 32/43

Supervaluationist solution How is the Sorite paradox solved? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 33/43

Supervaluationist solution How is the Sorite paradox solved? If P(last) is false in any precisification, then V 3 (P(last)) = 0; S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 33/43

Supervaluationist solution How is the Sorite paradox solved? If P(last) is false in any precisification, then V 3 (P(last)) = 0; If x or x is borderline case, then V 3 (P(x) P(x )) = = V 3 (P(x) P(x ))... Then V 2 (P(x) P(x )) = 1 in some model and W 2 (P(x) P(x )) = 0 in others! Thus, we cannot really say for any object in the Sorite series, that it is the last x such that x is P and x is not P! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 33/43

Supervaluationist solution How is the Sorite paradox solved? If P(last) is false in any precisification, then V 3 (P(last)) = 0; If x or x is borderline case, then V 3 (P(x) P(x )) = = V 3 (P(x) P(x ))... Then V 2 (P(x) P(x )) = 1 in some model and W 2 (P(x) P(x )) = 0 in others! Thus, we cannot really say for any object in the Sorite series, that it is the last x such that x is P and x is not P! we mistakenly conclude that P(x) P(x ) is true. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 33/43

Supervaluationist solution Pros: Accepts Meaning-Usage; Cons: 1. Usage is a fuzzy tri-partition, Meaning is a sharp tri-partition; 2. predicates cannot always precisified independently of one another; S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 34/43

Contextual solution A vague predicate depends on the context. Some implication P(k) P(k + 1) is false. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 35/43

Contextual solution A vague predicate depends on the context. Some implication P(k) P(k + 1) is false. We partially precisify P by classifying one of its borderline cases as a positive or a negative. The boundaries of P shift as the conversation proceed. It can been seen as a recursive tripartite approach... or a local supervaluation. How it works on a Sorite argument? P(k + 1) is false in the intended model, but we find it plausible because: 1. is not false before precisification, 2. assuming that some object x in the series is P, we change the model in which P(k) P(k + 1) holds. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 35/43

Contextual solution Pros: there is a more local and less idealised form of precisification; Cons: 1. what prompt the shift in the intended model? 2. the solution to the Sorite argument is not satisfactory for the speaker who believe that classifying a borderline case as P or not P is enough to render these classification true in some intended model: still, why P(k) P(k + 1) is true? A remark We said that context sensitivity is not vagueness... It is topic of debate whether fixing a context for the whole language (and not a comparison class) makes contextualism not appropriate for handling vagueness! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 36/43

Truth comes in degrees: fuzzy logic in vagueness From truth values set {0, 1} to a set D of degrees of truth. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 37/43

Truth comes in degrees: fuzzy logic in vagueness From truth values set {0, 1} to a set D of degrees of truth. Pros?? Cons?? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 37/43

Truth comes in degrees: fuzzy logic in vagueness From truth values set {0, 1} to a set D of degrees of truth. Pros?? Cons?? They depends on the structure of D, i.e. (linearly or partially) ordered, metric,... The most common approach is to take D = [0, 1], otherwise, we can require structural properties by taking D = D,,,,, 0, 1 as residuated lattice or an algebraic structure. About the Sorite: P(0) is completely true and P(1000000) is completely false. The premises P(k) P(k + 1) are very true, but not completely. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 37/43

Truth comes in degrees: fuzzy logic in vagueness Arguments for: 0. Degrees of truth account for vagueness as it is, without focusing on how it can be removed or reduced S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 38/43

Truth comes in degrees: fuzzy logic in vagueness Arguments for: 0. Degrees of truth account for vagueness as it is, without focusing on how it can be removed or reduced 1. Fuzzy theories can solve the Sorite paradox without any departure from the usual modus operandi of formal semantics! 2. Degrees of truth allow for a sharper definition of a vague predicate! What do I mean by sharper? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 38/43

Truth comes in degrees: fuzzy logic in vagueness Arguments for: 0. Degrees of truth account for vagueness as it is, without focusing on how it can be removed or reduced 1. Fuzzy theories can solve the Sorite paradox without any departure from the usual modus operandi of formal semantics! 2. Degrees of truth allow for a sharper definition of a vague predicate! What do I mean by sharper? What is water? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 38/43

Truth comes in degrees: fuzzy logic in vagueness Arguments for: 0. Degrees of truth account for vagueness as it is, without focusing on how it can be removed or reduced 1. Fuzzy theories can solve the Sorite paradox without any departure from the usual modus operandi of formal semantics! 2. Degrees of truth allow for a sharper definition of a vague predicate! What do I mean by sharper? What is water? A quite discussed proposal: vague predicates are tolerant. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 38/43

Truth comes in degrees: fuzzy logic in vagueness Arguments for: 0. Degrees of truth account for vagueness as it is, without focusing on how it can be removed or reduced 1. Fuzzy theories can solve the Sorite paradox without any departure from the usual modus operandi of formal semantics! 2. Degrees of truth allow for a sharper definition of a vague predicate! What do I mean by sharper? What is water? A quite discussed proposal: vague predicates are tolerant. Tolerance: if a and b are very similar in P-relevant aspects, then P(a) and P(b) are identical w.r.t. their truth. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 38/43

Truth comes in degrees But Tolerance creates contradictions: in a Sorite series, P(k) and P(k + 1) are all very similar, while P(0) is true and P(last) is false! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 39/43

Truth comes in degrees But Tolerance creates contradictions: in a Sorite series, P(k) and P(k + 1) are all very similar, while P(0) is true and P(last) is false! Another proposal [N. Smith] Vagueness is closeness. Closeness: if a and b are very similar in P-relevant aspects, then P(a) and P(b) are very similar w.r.t. their truth. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 39/43

Truth comes in degrees But Tolerance creates contradictions: in a Sorite series, P(k) and P(k + 1) are all very similar, while P(0) is true and P(last) is false! Another proposal [N. Smith] Vagueness is closeness. Closeness: if a and b are very similar in P-relevant aspects, then P(a) and P(b) are very similar w.r.t. their truth. Closeness allows to define vague predicates using degrees of truth and settles the Sorite series. Does it works for blurry boundaries and borderline cases? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 39/43

Truth comes in degrees Blurry boundaries. if a predicate P satisfies closeness, can it have sharp boundaries? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 40/43

Truth comes in degrees Blurry boundaries. if a predicate P satisfies closeness, can it have sharp boundaries? It can t! Otherwise we will have two objects that are very similar in P-relevant aspects, one of them true and the other false! Borderline cases. For a predicate that satisfies closeness, the existence of a Sorite series implies that some x in the extension of P is neither true or false: that is, a borderline case! S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 40/43

Truth comes in degrees Arguments against: Artificial precision. How can we assess that a coat is red with degree 0.897 instead of 0.899? S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 41/43

Truth comes in degrees Arguments against: Artificial precision. How can we assess that a coat is red with degree 0.897 instead of 0.899? This problem only applies to those theories that either use [0, 1] as set of truth values or fix a unique intended model for each vague argument. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 41/43

Truth comes in degrees Arguments against: Artificial precision. How can we assess that a coat is red with degree 0.897 instead of 0.899? This problem only applies to those theories that either use [0, 1] as set of truth values or fix a unique intended model for each vague argument. There are some solution in literature, one of them is fuzzy plurivaluationism: from one unique model to multiple acceptable models. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 41/43

Truth comes in degrees Arguments against: Truth functionality. Truth values are incompatible with "ordinary" use of compound propositions when dealing with borderline cases. One example is the fact that "x is Red" or "x is not red" is not always true. But these arguments forget about the wide plethora of fuzzy logics and they seem to be always argued only based on the critics s intuitions. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 42/43

Truth comes in degrees Arguments against: Truth functionality. Truth values are incompatible with "ordinary" use of compound propositions when dealing with borderline cases. One example is the fact that "x is Red" or "x is not red" is not always true. But these arguments forget about the wide plethora of fuzzy logics and they seem to be always argued only based on the critics s intuitions. There are just as many arguments that confute this critics. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 42/43

Truth comes in degrees Arguments against: Truth functionality. Truth values are incompatible with "ordinary" use of compound propositions when dealing with borderline cases. One example is the fact that "x is Red" or "x is not red" is not always true. But these arguments forget about the wide plethora of fuzzy logics and they seem to be always argued only based on the critics s intuitions. There are just as many arguments that confute this critics. Empirical data can be found in order to lean both in favor and against truth functionality of fuzzy logics. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 42/43

End of Lecture 1. S. Lapenta and D. Valota (ESSLLI 2018) Lecture 1 43/43