September 12th, 2013

SuB18 (Poster Appendix)

Universal Quantification and Degree Modification as Slack Regulation Heather Burnett (Universit´e de Montr´eal & Institut Jean Nicod) 1. The Delineation Tolerant, Classical, Strict (DelTCS) Framework (Burnett, 2012). 2. Mereological Extension of DelTCS (Burnett, 2013).

1

Delineation Tolerant, Classical, Strict

1.1

The Language

The vocabulary consists of the following expressions: 1. A series of individual constants: a1 , a2 , a3 . . . 2. A series of individual variables: x1 , x2 , x3 . . . 3. Three series of unary predicate symbols: • Relative scalar adjectives: P, P1 , P2 , P3 . . . • Total absolute scalar adjectives: Q, Q1 , Q2 , Q3 . . . • Partial absolute scalar adjectives: R, R1 , R2 , R3 . . . 4. For every unary predicate symbol P , there is a binary predicate >P . 5. For every unary predicate symbol P , there is a binary predicate ∼P . 6. Quantifier ∀ and connectives ∧, ∨, ¬ and →, plus parentheses. The syntax of DelTCS is as follows: 1. Variables and constants (and nothing else) are terms. 2. If t is a term and P is a predicate symbol, then P (t) is a well-formed formula (wff). 3. If t1 and t2 are terms and P is a predicate symbol, then t1 >P t2 is a wff. 4. If t1 and t2 are terms and P is a predicate symbol, then t1 ∼P t2 is a wff. 5. For any variable x, if φ and ψ are wffs, then ¬φ, φ ∧ ψ, φ ∨ ψ, φ → ψ, and ∀xφ are wffs. 6. Nothing else is a wff.

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1.2

SuB18 (Poster Appendix)

The Semantics

We start with a basic model (called a C(lassical) model in the terminology of Cobreros et al. (2012)) in which the interpretation of predicates is relativized to comparison classes as follows: Definition 1.1 C-model. A c-model is a tuple M = hD, mi where D is a non-empty domain of individuals, and m is a function from pairs consisting of a member of the non-logical vocabulary and a comparison class (a subset of the domain) satisfying: • For each individual constant a1 , m(a1 ) ∈ D. • For each X ⊆ D and for each predicate P , m(P, X) ⊆ X. In order to incorporate the additional structure of TCS into our delineation system, we extend the C-model (defined in definition 1.1) into a T(olerant) model by adding indifference relations (∼P s) into the model structure as follows: Definition 1.2 T-model. A t-model is a tuple M = hD, m, ∼i, where hD, mi is a c-model and ∼ is a function from predicate-comparison class pairs such that: • For all P and all X ⊆ D, ∼X P is a binary relation on X. I will discuss what appropriate constraints should be put on the indifference relations below. The interpretation of variables is done in the standard way (on assignment): Definition 1.3 Assignment. An assignment for a c/t-model M is a function g : {xn : n ∈ N} → D (from the set of variables to the domain D). Formulas are interpreted in three ways: classically, tolerantly and strictly. The classical interpretation of terms is given as follows: Definition 1.4 Interpretation of terms (J·KM,g ). For a model M , an assignment g, 1. If x1 is a variable, Jx1 KM,g = g(x1 ). 2. If a1 is a constant, Ja1 KM,g = m(a1 ). In what follows, for an interpretation J·KM,g , a variable x1 , and a constant a1 , let g[a1 /x1 ] be the assignment for M which maps x1 to a1 , but agrees with g on all variables that are distinct from x1 . In definition 1.5, formulas are interpreted relative to a model, an assignment function, and a comparison class; however, in what follows, for readability considerations, I will often omit the model and assignment notation, writing only J·KX for J·KM,g,x . Definition 1.5 Classical Satisfaction (J·K). For all interpretations J·KM,g , all X ∈ P(D), all formulas φ, ψ, all predicates P , and all terms t1 , t2 ,   1 if Jt1 KM,g ∈ m(P, X) 1. JP (t1 )KM,g,X = 0 if Jt1 KM,g ∈ X − m(P, X)   i otherwise 2

September 12th, 2013

2. Jt1 >P t2 KM,g,X 3. Jt1 ∼P t2 KM,g,X

4.

5.

6.

7.

8.

SuB18 (Poster Appendix)

( 1 if there is some X 0 ⊆ D : JP (t1 )KM,g,X 0 = 1 and JP (t2 )KM,g,X 0 = 0 = 0 otherwise ( 1 if t1 ∼X P t2 = 0 otherwise

  1 if JφKM,g,X = 0 J¬φKM,g,X = 0 if JφKM,g,X = 1   i otherwise   1 if JφKM,g,X = 1 and JψKM,g,X = 1 Jφ ∧ ψKM,g,X = 0 if {JφKM,g,X , JψKM,g,X } = {1, 0} or {0}   i otherwise   1 if JφKM,g,X = 1 or JψKM,g,X = 1 Jφ ∨ ψKM,g,X = 0 if JφKM,g,X = JψKM,g,X = 0   i otherwise   1 if JφKM,g,X = 0 or JψKM,g,X = 1 Jφ → ψKM,g,X = 0 if JφKM,g,X = 1 and JψKM,g,X = 0   i otherwise   1 if for every a1 ∈ X, JφKM,g[a1 /x1 ],X = 1 J∀x1 φKM,g,X = 0 if for some a1 ∈ X, JφKM,g[a1 /x1 ],X = 0   i otherwise

In this work, I will adopt the set of constraints on the application of gradable predicates presented in van Benthem (1982) and van Benthem (1990)1 . Van Benthem proposes three axioms governing the behaviour of individuals across comparison classes. They are the following (presented in my notation): For all predicates P1 , all interpretations J·KM,g , all X ⊆ D and a1 , a2 ∈ X such that JP1 (a1 )KM,g,X = 1 and JP1 (a2 )KM,g,X = 0, 1. No Reversal (NR:) There is no X 0 ⊆ D such that JP1 (a2 )KM,g,X 0 = 1 and JP (a1 )KM,g,X 0 = 0. 2. Upward difference (UD): For all X 0 ⊆ D, if X ⊆ X 0 , then there are some a3 , a4 : JP1 (a3 )KM,g,X 0 = 1 and JP1 (a4 )KM,g,X 0 = 0. 3. Downward difference (DD): For all X 0 ⊆ D, if X 0 ⊆ X and a1 , a2 ∈ X 0 , then there are some a3 , a4 : JP1 (a3 )KM,g,X 0 = 1 and JP1 (a4 )KM,g,X 0 = 0. 1

Although see van Rooij (2011) for another set of constraints that could also be adopted in this framework.

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September 12th, 2013

SuB18 (Poster Appendix)

Theorem 1.1 (van Benthem, 1982) shows that orders lexicalized by the comparative form of gradable predicates can be constructed from the context-sensitivity of the positive form and certain axioms governing the application of the predicate across different contexts. Theorem 1.1 Strict Weak Order. For all P , >P is a (possibly) strict weak order with at least two ‘degrees’. I propose that a different axiom set governs the semantic interpretation of the members of the absolute class that does not apply to the relative class: the singleton set containing the absolute adjective axiom. (1)

Absolute Adjective Axiom (AAA): For all total and partial predicates Q1 , all interpretations J·KM,g , all X ∈ P(D) and a1 ∈ X, 1.If JQ1 (a1 )KM,g,X = 1, then JQ1 (a1 )KM,g,D = 1.

2.If JQ1 (a1 )KM,g,D = 1, and JQ1 (a1 )KM,g,X 6= i, then JQ1 (a1 )KM,g,X = 1. Consequence: the relations denoted by the absolute and non-scalar comparative (>Q ) do not allow for the predicate to distinguish three distinct individuals. Theorem 1.2 If Q satisfies the AAA, then there is no model M such that, for distinct x, y, z ∈ D, x >Q y >Q z. 1.2.1

Tolerant/Strict Interpretations and Vagueness

As in TCS, the tolerant and strict interpretations are interdefined based on the classical interpretation and the predicate-relative indifference relations, with J·Kt and J·Ks being duals. Definition 1.6 Tolerant Satisfaction(J·Kt ). For all interpretations J·KM,g , all X ⊆ D, all formulas φ, ψ, all predicates P , and all terms t1 , t2 ,  X  1 if there is some a1 ∼P Jt1 KM,g : JP (a1 )KM,g,X = 1 1. JP (t1 )KtM,g,X = 0 if Jt1 KM,g ∈ X, and there is no a1 ∈ X : a1 ∼X P JP (t1 )KM,g = 1   i otherwise ( 1 if there is some X 0 ⊆ D : JP (t1 )KtM,g,X 0 = 1 and JP (t2 )KtM,g,X 0 = 0 2. Jt1 >P t2 KtM,g,X = 0 otherwise ( 1 if t1 ∼X P t2 3. Jt1 ∼P t2 KtM,g,X = 0 otherwise  s  1 if JφKM,g,X = 0 4. J¬φKtM,g,X = 0 if JφKsM,g,X = 1   i otherwise  t t  1 if JφKM,g,X = 1 and JψKM,g,X = 1 5. Jφ ∧ ψKtM,g,X = 0 if {JφKtM,g,X , JψKtM,g,X } = {1, 0} or {0}   i otherwise 4

September 12th, 2013

SuB18 (Poster Appendix)

 t t  1 if JφKM,g,X = 1 or JψKM,g,X = 1 6. Jφ ∨ ψKtM,g,X = 0 if JφKtM,g,X = JψKtM,g,X = 0   i otherwise  t t  1 if JφKM,g,X = 0 or JψKM,g,X = 1 7. Jφ → ψKtM,g,X = 0 if JφKtM,g,X = 1 and JψKtM,g,X = 0   i otherwise  t  1 if for every a1 ∈ X, JφKM,g[a1 /x1 ],X = 1 8. J∀x1 φKtM,g,X = 0 if for some a1 ∈ X, JφKtM,g[a1 /x1 ],X = 0   i otherwise Definition 1.7 Strict Satisfaction(JKs ). For all interpretations J·KM,g , all X ⊆ D, all formulas φ, ψ, all predicates P , and all terms t1 , t2 ,  X  1 if for all a1 ∼P Jt1 KM,g : JP (a1 )KM,g,X = 1 1. JP (t1 )KsM,g,X = 0 if Jt1 KM,g ∈ X, and there is no a1 ∈ X : a1 ∼X P JP (t1 )KM,g = 1   i otherwise ( 1 if there is some X 0 ⊆ D : JP (t1 )KsM,g,X 0 = 1 and JP (t2 )KsM,g,X 0 = 0 2. Jt1 >P t2 KsM,g,X = 0 otherwise ( 1 if t1 ∼X P t2 3. Jt1 ∼P t2 KsM,g,X = 0 otherwise  t  1 if JφKM,g,X = 0 4. J¬φKsM,g,X = 0 if JφKtM,g,X = 1   i otherwise  s s  1 if JφKM,g,X = 1 and JψKM,g,X = 1 5. Jφ ∧ ψKsM,g,X = 0 if {JφKsM,g,X , JψKsM,g,X } = {1, 0} or {0}   i otherwise  s s  1 if JφKM,g,X = 1 or JψKM,g,X = 1 6. Jφ ∨ ψKsM,g,X = 0 if JφKtM,g,X = JψKtM,g,X = 0   i otherwise  s s  1 if JφKM,g,X = 0 or JψKM,g,X = 1 7. Jφ → ψKsM,g,X = 0 if JφKsM,g,X = 1 and JψKsM,g,X = 0   i otherwise  s  1 if for every a1 ∈ X, JφKM,g[a1 /x1 ],X = 1 8. J∀x1 φKsM,g,X = 0 if for some a1 ∈ X, JφKsM,g[a1 /x1 ],X = 0   i otherwise 5

September 12th, 2013

SuB18 (Poster Appendix)

Constraints on ∼P relations: (2)

Reflexivity (R): For all predicates P , all interpretations J·KM,g , all X ⊆ D, for all a1 ∈ X, a1 ∼X P 1 a1 .

(3)

(Non)Symmetry in ∼: 1.Symmetry (S): For a relative predicate P1 , an interpretation J·KM,g , and X a1 , a2 ∈ D, if a1 ∼X P1 a2 , then a2 ∼P1 a1 . 2.Total Axiom (TA): For a total predicate Q1 , an interpretation J·KM,g , and a1 , a2 ∈ D, if JQ1 (a1 )KM,g,D = 1 and JQ1 (a2 )KM,g,D = 0, then a2 6∼X Q1 a1 , for all X ⊆ D. 3.Partial Axiom (PA): For a partial predicate R1 , an interpretation J·KM,g and a1 , a2 ∈ D, if JR1 (a1 )KM,g,D = 1 and JR1 (a2 )KM,g,D = 0, then a1 6∼X R 1 a2 , for all X ⊆ D.

The rest of the constraints on indifference relations across comparison classes that I will propose apply to all classes of predicates in the same way2 . Tolerant Convexity: For all predicates P1 , all interpretations J·KM,g , all X ⊆ D, and all a1 , a2 ∈ X,

(4)

t t •If a1 ∼X P1 a2 and there is some a3 ∈ X such that a1 ≥P1 a3 ≥P1 a2 , then a1 ∼X P 1 a3 .

Strict Convexity: For all predicates P1 , all interpretations J·KM,g , all X ⊆ D, and all a1 , a2 ∈ X,

(5)

s s •If a1 ∼X P1 a2 and there is some a3 ∈ X such that a1 ≥P1 a3 ≥P1 a2 , then a3 ∼X P 1 a2 .

Granularity (G): For all predicates P1 , all interpretations J·KM,g , all X ⊆ D, and all a1 , a2 ∈ X,

(6)

0

0 0 X •If a1 ∼X P1 a2 , then for all X ∈ P(D) : X ⊆ X , a1 ∼P1 a2 .

For all predicates P1 , all interpretations J·KM,g , all X ⊆ D,

(7) 2

The definition of ≥P that is featured in the next two constraints is as follows: We first define an equivalence relation ≈P : Definition 1.8 Equivalent. (≈) For an interpretation J·KM,g,X , a predicate P , a1 , a2 ∈ D: 1. a1 ≈P a2 iff Ja1 >P a2 KM,g,X = 0 and Ja2 >P a1 KM,g,X = 0.

2. a1 ≈tP a2 iff Ja1 >P a2 KtM,g,X = 0 and Ja2 >P a1 KtM,g,X = 0.

3. a1 ≈sP a2 iff Ja1 >P a2 KsM,g,X = 0 and Ja2 >P a1 KsM,g,X = 0. Now we define ≥P :

Definition 1.9 Greater than or equal. (≥) For an interpretation J·KM,g,X , a predicate P , a1 , a2 ∈ D: 1. a1 ≥P a2 iff Ja1 >P a2 KM,g,X = 1 or a1 ≈P a2 .

2. a1 ≥tP a2 iff Ja1 >P a2 KtM,g,X = 1 or a1 ≈tP a2 .

3. a1 ≥sP a2 iff Ja1 >P a2 KsM,g,X = 1 or a1 ≈sP a2 .

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SuB18 (Poster Appendix)

1.Contrast Preservation (CP): For all X 0 ⊆ D, and a1 , a2 ∈ X, if X ⊂ X 0 X0 0 X0 and a1 6∼X P1 a2 and a1 ∼P1 a2 , then ∃a3 ∈ X − X : a1 6∼P1 a3 . 2.Minimal Difference (MD): For all a1 , a2 ∈ D, if Ja1 >P1 a2 KM,g,X = 1, {x,y} then a1 6∼P1 a2 . In summary, I propose that the definitions of the ∼P s, ∼Q s and ∼R s are constrained by the axioms in table 1. Constraint Relative (∼P ) Total (∼Q ) Reflexivity (R) X X Symmetry (S) X × Total Axiom (TA) × X Partial Axiom (PA) × × Tolerant Convexity (TC) X X Strict Convexity (SC) X X Granularity (G) X X Minimal Difference (MD) X X Contrast Preservation (CP) X X

Partial (∼R ) X × × X X X X X X

Table 1: Constraints on the definition of ∼ for relative, total and partial predicates

1.3 1.3.1

(Pertinent) Results Definition of Scale Structure Classes

Definition 1.10 Top-closed scale. For a predicate P in a model M , >P is an topclosed scale iff for all extensions of M, M 0 , there is no a1 ∈ DM 0 − DM such that a1 >P a2 in M 0 , for a2 : ¬∃a3 : a3 >P a2 in M . Definition 1.11 Bottom-closed scale. For a predicate P in a model M , >P is an bottom-closed scale iff for all extensions of M, M 0 , there is no a1 ∈ DM 0 − DM such that a2 >P a1 in M 0 , for a2 : ¬∃a3 : a2 >P a3 in M . Definition 1.12 Open Scale. For a predicate P in a model M , >P is an open scale iff >P is neither top-closed nor bottom-closed in M . 1.3.2

Scale Structure Results

Theorem 1.3 If Q is a total absolute adjective (i.e. subject to the AAA and the relevant constraints in table 1), tQ is a top-closed scale. Theorem 1.6 If Q ∈ AA, then >sQ is a bottom closed scale. Theorem 1.7 If P is a relative adjective, then >P is an open scale. 7

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SuB18 (Poster Appendix)

Connection between strict interpretations and scale structure for total adjectives: Theorem 1.8 Total Top Endpoint. If Q is an absolute adjective (i.e. satisfies the AAA), all models M , and a2 ∈ D, • If JQ(a2 )KD = 1 then there is no a3 ∈ D such that a3 >tQ a2 .

2

Mereological Extension of DelTCS (Burnett, 2013)

The language is the following:

2.1

Vocabulary

1. A series of singular constants: a1 , a2 , a3 . . . 2. A series of group constants: g1 , g2 , g3 . . . 3. For every group constant, there is an individual second-order predicate: Ig1 , Ig2 , Ig3 . . . 4. Four series of unary predicate symbols: • Singular predicates: P, P1 , P2 , P3 . . . • Distributive predicates: Q, Q1 , Q2 , Q3 . . . • Complex collective predicates: R, R1 , R2 , R3 . . . • Atomic collective predicates: S, S1 , S2 , S3 . . . 5. There is a binary predicate . 6. For every individual Ig1 , there is a binary predicate >Ig1 . 7. For every individual Ig1 , there is a binary predicate ∼Ig1 .

2.2

Syntax

1. If a1 is a singular constant and P1 is a singular predicate symbol, then P1 (a1 ) is a well-formed formula (wff). 2. If g1 is a group constant and Q is a distributive, complex collective or atomic collective predicate, then Q(g1 ) is a wff. 3. If Ig1 is an individual and P is a predicate symbol, then Ig1 (P ) is a wff. 4. If g1 and g2 are group constants, then g1  g2 is a wff. 5. If Q1 and Q2 are distributive, complex collective or atomic collective predicate symbols and Ig1 is an individual, then Q1 >Ig1 Q2 is a wff. 6. If Q1 and Q2 are distributive, complex collective or atomic collective predicate symbols and Ig1 is an individual, then Q1 ∼Ig1 Q2 is a wff. 7. Nothing else is a wff. 8

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2.3

Basic Semantics

2.3.1

Model Structure

SuB18 (Poster Appendix)

Definition 2.1 A model structure M is a tuple hD, , ∼i such that 1. D is a set of objects. 2.  is a partial order on D (a reflexive, transitive and anti-symmetric relation). 3. ∼ is a two place function such that, for every individual Ig1 and every family of sets X ⊆ P(D), ∼ (Ig1 , X) is a binary relation (notated ∼X Ig1 ) on the elements of X. • We will place more constraints on the definition of the ∼X Ig1 s below. 2.3.2

Constraints on 

In line with much work in the formal semantics of determiner phrases and plural predicates (since Link (1983)), I assume certain constraints on the  relation such as hD, i forms a classical extensional mereology. • Following Hovda (2008), we adopt the constraints shown below (defined using the following definitions). Definition 2.2 Overlap (◦). For all g1 , g2 ∈ D, g1 ◦ g2 iff ∃g3 ∈ D such that g3  g1 and g3  g2 . Definition 2.3 Fusion (Fu). For g1 ∈ D and X ⊆ D, Fu(g1 , X) (‘g1 fuses X’) iff, for all g2 ∈ D, (8)

g2 ◦ g1 iff there is some g3 such that g3 ∈ X and g2 ◦ g3 .

We now adopt the following constraints on hD, i: 1. Reflexivity. For all g1 ∈ D, g1  g1 . 2. Transitivity. For all g1 , g2 , g3 , if g1  g2 and g2  g3 , then g1  g3 . 3. Anti-symmetry. For all g1 , g2 ∈ D, if g1  g2 and g2  g1 , then g1 = g2 . 4. Strong Supplementation. For all g1 , g2 ∈ D, for all g3 , if, if g3  g1 , then g3 ◦ g2 , then g1  g2 . 5. Fusion Existence. For all X ⊆ D, if there is some g1 ∈ X, then there is some g2 ∈ D such that F u(g2 , X). We can note that, in CEM, for every subset of D, not only does its fusion exist, but it is also unique (cf. Hovda 2009, p. 70). Therefore, in what follows, I will often use the following notation: W W Definition 2.4 Join/sum/fusion ( ). For all X ⊆ D, X is the unique g1 such that F u(g1 , X). W • Occasionally, we will write g1 + g2 for {g1 , g2 }. Finally, since D is finite, the CEMs that we are interested in are atomic. The property of being an atom is defined below3 . 3

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Definition 2.6 Atoms. Let P ⊆ D. The atoms of P (notated AT (P )) is the set {g1 : g1 ∈ P & this is no g2 such that g2 ≺ g1 }.

2.4

Model and Interpretation

Definition 2.7 A model is a tuple hD, , ∼, J·Ki such that hD, , ∼i is a model structure (as defined above) and J·K is a function from elements of the non-logical vocabulary and comparison classes X ⊆ P(D) such that: 1. If a1 is a singular constant, then Ja1 KX ∈ AT (D). 2. If g1 is a group constant, then Jg1 KX ∈ D. 3. If Ig1 is an individual, then JIg1 KX ⊆ X. 4. If P1 is a singular predicate, then JP1 KX ⊆ AT (D). 5. If Q2 is a distributive, complex collective or atomic collective predicate, then JQ2 KX ⊆ D. 2.4.1

Constraints on J·K

The interpretation of certain elements of the non-logical vocabulary is further constrained by certain axioms. • Firstly, the interpretation of the Ig1 s is not arbitrary and free to vary across comparison classes. Rather, it is tied to the interpretation of the g1 s in the way proposed by Montague and shown in (9). (9)

Montagovian Individuals: For all Ig1 and comparison classes X ⊆ P(D), JIg1 KX = {Q : g1 ∈ Q}. • Secondly, the different kinds of plural predicates have different mereological structures. 1. Distributive predicates are, themselves, classical atomic extensional mereologies. (10)

Distributivity: For all distributive predicates Q, all groups g1 ∈ D and all comparison classes X, g1 ∈ JQKX iff g1 = g2 + g3 , for g2 , g3 ∈ JQKX .

2. Complex collective predicates have no atomic members. (11)

Collectivity: For all collective predicates R, groups g1 ∈ D, and comparison classes X ⊆ P(D), if g1 ∈ JRKX , then g1 ∈ / AT (D).

3. Atomic collective predicates have not atomic members and, furthermore, if they hold of a group, then they do not hold of any subgroup. Definition 2.5 Proper part(≺). For all g1 , g2 ∈ D, g1 ≺ g2 iff g1  g2 and g1 6= g2 .

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(12)

SuB18 (Poster Appendix)

Atomicity: For all atomic collective predicates S, groups g1 ∈ D, and comparison classes X ⊆ P(D), if g1 ∈ JSKX , then there is no g2 ∈ JSKX such that g2 ≺ g1 .

Axiom Distributive Q Complex Coll. R Distributivity (D) X × Collectivity (C) × X Atomicity (A) × ×

Atomic Coll. S × X X

Table 2: Constraints on the semantic denotation of plural predicates.

2.4.2

Interpretation

Definition 2.8 Classical Satisfaction (J·K). For all models M , all comparison classes X ⊆ P(D), all a1 ∈ AT (D), all g1 , g2 ∈ D, singular predicates P and plural predicates Q1 , Q2 , ( 1 if Ja1 KM,X ∈ JP KM,X 1. JP (a1 )KM,X = 0 otherwise ( 1 if Jg1 KM,X ∈ JQ1 KM,X 2. JQ1 (g1 )KM,X = 0 otherwise   1 if JQ1 KM,X ∈ JIg1 KM,X 3. JIg1 (Q1 )KM,X = 0 if JQ1 KM,X ∈ X − JIg1 KM,X   i otherwise ( 1 if g1  g2 4. Jg1  g2 KM,X = 0 otherwise  0  1 if there is some X ⊆ P(D) : JIg1 (Q1 )KX,M = 1 5. JQ1 >Ig1 Q2 KM,X = and JIg1 (Q2 )KX,M = 0   0 otherwise ( 1 if Q1 ∼X Ig1 Q2 6. JQ1 ∼Ig1 Q2 KM,X = 0 otherwise

2.5

Classical Semantics Results

• With the constraint in (9), we can immediately show that our individual (i.e. definite plural DP) predicates satisfy a DP version of the AAA. Theorem 2.1 Absolute Adjective Theorem. Let Ig1 be an individual. Show that for all X ⊆ P(D) and all predicates Q ∈ X, 1. If JIg1 (Q)KX = 1 then JIg1 (Q)KP(D) = 1. 11

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2. If JIg1 (Q)KP(D) = 1 and JIg1 (Q)KX is not indefinite, then JIg1 (Q)KX = 1. • Furthermore, it immediately follows that the scale associated with the semantic denotation of Ig1 (>Ig1 ) has at most 2 degrees. Theorem 2.2 For all individuals Ig1 , there is no model M such that, for distinct Q1 , Q2 , Q3 ⊆ D, Q1 >Ig1 Q2 >Ig1 Q3 .

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Tolerant and Strict Semantics

Definition 3.1 Tolerant/Strict Interpretation of Individuals. For all groups g1 ∈ D and comparison classes X, 1. JIg1 KtX = {Q1 : there is some Q2 ∼X Ig1 Q1 : JIg1 (Q2 )KX = 1 2. JIg1 KsX = {Q1 : for all Q2 : Q2 ∼X Ig1 Q1 : JIg1 (Q2 )KX = 1 Definition 3.2 Tolerant Satisfaction (J·Kt ). For all models M , all comparison classes X ⊆ P(D), all a1 ∈ AT (D), all g1 , g2 ∈ D, singular predicates P and plural predicates Q1 , Q2 , ( 1 if Ja1 KM,X ∈ JP KM,X 1. JP (a1 )KtM,X = 0 otherwise ( 1 if Jg1 KM,X ∈ JQ1 KM,X 2. JQ1 (g1 )KtM,X = 0 otherwise  t  1 if JQ1 KM,X ∈ JIg1 KM,X 3. JIg1 (Q1 )KtM,X = 0 if JQ1 KM,X ∈ X − JIg1 KtM,X   i otherwise ( 1 if g1  g2 4. Jg1  g2 KtM,X = 0 otherwise  0 t  1 if there is some X ⊆ P(D) : JIg1 (Q1 )KX,M = 1 5. JQ1 >Ig1 Q2 KtM,X = and JIg1 (Q2 )KtX,M = 0   0 otherwise ( 1 if Q1 ∼X Ig1 Q2 t 6. JQ1 ∼Ig1 Q2 KM,X = 0 otherwise Definition 3.3 Strict Satisfaction (J·Ks ). For all models M , all comparison classes X ⊆ P(D), all a1 ∈ AT (D), all g1 , g2 ∈ D, singular predicates P and plural predicates Q1 , Q2 , ( 1 if Ja1 KM,X ∈ JP KM,X 1. JP (a1 )KsM,X = 0 otherwise 12

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SuB18 (Poster Appendix)

( 1 2. JQ1 (g1 )KsM,X = 0   1 s 3. JIg1 (Q1 )KM,X = 0   i ( 1 4. Jg1  g2 KsM,X = 0

5. JQ1 >Ig1 Q2 KsM,X

6. JQ1 ∼Ig1 Q2 KtM,X

3.1 (13)

if Jg1 KM,X ∈ JQ1 KM,X otherwise if JQ1 KM,X ∈ JIg1 KsM,X if JQ1 KM,X ∈ X − JIg1 KsM,X otherwise

if g1  g2 otherwise  0 s  1 if there is some X ⊆ P(D) : JIg1 (Q1 )KX,M = 1 = and JIg1 (Q2 )KsX,M = 0   0 otherwise ( 1 if Q1 ∼X Ig1 Q2 = 0 otherwise

Constraints on ∼ Shared parts (SP): For all models M , groups g1 ∈ D, comparison classes X ⊆ P(D), and distinct properties Q1 , Q2 ∈ X, •If Q1 ∼X Ig1 Q2 , then there is some g2  g1 such that g2 ∈ Q1 and g2 ∈ Q2 .

Definition 3.4 Upper bounds (↑g1 ). For all properties Q1 and g1 ∈ D, ↑g1 (Q1 ) = {g2 : g1 ≤ g2 , and there is no g3 : g3 ∈ Q1 and g1 > g3 > g2 }. (14)

Mereological convexity (MC): For all models M , groups g1 ∈ D, comparison classes X ⊆ P(D), and properties Q1 , Q2 ∈ X, •If Q1 ∼X Ig1 Q2 and there is some Q3 ∈ X such that g2  g3  g4 , for some g2 ∈↑g1 (Q1 ), g3 ∈↑g1 (Q3 ) and g4 ∈↑g1 (Q4 ), then Q1 ∼X Ig1 Q3 .

(15)

Incomparability (I): For all models M , groups g1 ∈ D, comparison classes X ⊆ P(D), and properties Q1 ∈ X, for all Q2 ⊆ D, •If there is some g2 ∈↑g1 (Q2 ) such that g2 is incomparable to g3 , for some g3 ∈↑g1 (Q1 ), then Q2 ∈ / X.

(16)

Granularity (G): For all individuals Ig1 , all interpretations J·KM,g , all X ⊆ P(D), and all Q1 , Q2 ∈ X, 0

0 0 X •If Q1 ∼X Ig1 Q2 , then for all X ⊆ P(D) : X ⊆ X , Q1 ∼Ig1 Q2 .

(17)

For all individuals Ig1 , all interpretations J·KM,g , all X ⊆ P(D), a. Contrast Preservation (CP): For all X 0 ⊆ P(D), and Q1 , Q2 ∈ X, if X0 0 X0 X ⊂ X 0 and Q1 6∼X Ig1 Q2 and Q1 ∼Ig1 Q2 , then ∃Q3 ∈ X − X : Q1 6∼Ig1 Q3 . b. Minimal difference (MD): For all Q1 , Q2 ∈ P(D), if JQ1 >Ig1 Q2 KM,g,X = {Q ,Q } 1, then Q1 6∼Ig11 2 Q2 .

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SuB18 (Poster Appendix)

Tolerant Scale Results (>t) • We can first show that the >t relations are weak orders (i.e. antisymmetric and transitive).

Theorem 4.1 Asymmetry. For all models M , all groups g1 ∈ D, all properties Q1 , Q2 ⊆ D, if Q1 >tIg1 Q2 , then Q2 6>tIg1 Q1 . Theorem 4.2 Transitivity. For all models M , all groups g1 ∈ D, all properties Q1 , Q2 , Q3 ⊆ D, if Q1 >tIg1 Q2 and Q2 >tIg1 Q3 , then Q1 >tIg1 Q3 . • For similar reasons to total AAs above, since Ig1 satisfies the Absolute Adjective Theorem, the >t have a top-endpoint.

References Burnett, H. (2012). The Grammar of Tolerance: On Vagueness, Context-Sensitivity, and the Origin of Scale Structure. PhD thesis, University of California, Los Angeles. Burnett, H. (2013). The Logical Foundations of Gradability in Natural Language. Under review for publication by Oxford University Press. ´ e, P., Ripley, D., and van Rooij, R. (2012). Tolerant, classical, strict. Cobreros, P., Egr´ Journal of Philosophical Logic, 41:347–385. Hovda, P. (2008). What is classical mereology? Journal of philosophical logic, 38:55–82. Link, G. (1983). The logical analysis of plurals and mass nouns: A lattice-theoretic approach. In Bauerle, R., Schwartze, C., and von Stechow, A., editors, Meaning, Use and the interpretation of language, pages 302–322. Mouton de Gruyter, The Hague. van Benthem, J. (1982). Later than late: On the logical origin of the temporal order. Pacific Philosophical Quarterly, 63:193–203. van Benthem, J. (1990). The logic of time. Reidel, Dordrecht. ´ e, P. and Klinedinst, N., van Rooij, R. (2011). Implicit vs explicit comparatives. In Egr´ editors, Vagueness and Language Use, pages –. Palgrave Macmillan.

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