FSP 2012 – GRADABILITY. Concept Properties like height, cleanliness and fullness are called Gradable since they may manifest to different extents over

FSP 2012 – GRADABILITY

Concept• Properties like height, cleanliness and fullness are called

Gradable since they may manifest to different extents over time/place.

• These properties are captured in language by adjectives: tall/short, clean/dirty and full/empty. Hence: Gradable Adjectives.

• Gradability investigates the way in which gradable adjectives capture gradable properties and aims at formalizing the characteristics of these adjectives.

Linguistic Evidence• Markers of Gradability

• The Comparative Morpheme• John is taller/shorter than Bill• The red towel is wetter/drier than the blue one• My glass is fuller/emptier than yours

• Degree Modifiers• John is very tall/short• The towel is slightly wet / almost dry• My glass is completely full/empty

Non-Gradable Adjectives• Not all adjectives are gradable:

• #John is more married than Bill• #My door is more hand-made than Mary’s• #The wall is more stony than the floor

Semantic Analysis of Gradable Adjectives

• Consider the sentence:• Yoni is taller than Danny

Yoni

Danny



• A common analysis of the semantics of gradable adjectives assumes that:• Gradable adjectives are functions that map their arguments onto

abstract representations of measurement called Degrees, associated with the primitive type d. In this analysis, gradable adjectives are called Measure Functions.

• A set of degrees totally ordered with respect to some dimension (height, fullness, etc.) constitutes a Scale.



YoniDanny

174 1940

The scale of tall/short

Degrees of height (in cm)

Isaac

185

Gidi

176

Compositional Analysis• Consider the sentence:

• Yoni is taller than Danny

• We adopt a degree-basedapproach by which gradable adjectives denote functions from individuals to degrees, hence type ed

Scale Structure• Consider the following data:

• John is #slightly/#completely tall/short• The table is slightly/#completely dirty• The room is #slightly/completely clean• The gas tank is slightly/completely full/empty

• These sentences indicate that the acceptability of degree modifiers depends on the adjective being modified.

• We can use this data to hypothesize about the scale structure of each adjective.

• Assuming that slightly picks a (near-) minimum degree value and completely picks a maximum value, we conclude a typology of adjectives based on their scale structure.

Typology of Scales• Adjectives with open scales, e.g. tall, short:

slightly completely

• Adjectives with lower-closed scales, e.g. dirty, wet:

slightly completely

• Adjectives with upper-closed scales, e.g. clean, dry:

slightly completely

• Adjectives with totally-closed scales, e.g. full, empty:

slightly completely

Distributional Puzzle #1 – ‘Almost’• Consider the following data:

• John is almost #tall/#short tall short• The table is almost #dirty/clean dirty clean• The gas tank is almost full/empty full empty

• These sentences indicate that the acceptability of ‘almost’ depends on the adjective being modified.

• Answer the following questions:• Characterize the adjectives that can and cannot be modified by

‘almost’ based on their scale structure.

• Illustrate this characterization with the adjectives: deep/shallow, wet/dry and certain/uncertain.

The Positive Form of Gradable Adjectives

• Consider the sentence: Yoni is tall



• What are the requirements from Yoni to be counted as tall?




• Where these requirements are coming from?




• Where these requirements are coming from?

• How the meaning can be calculated compositionality?


• Consider the sentence: Yoni is tall• Intuitively, Yoni needs to exceed some degree of height in

order to count as tall. This degree is called the Standard.• The average of all relevant entities in the context of

utterance is a natural choice for a standard.

• It follows that in one context Yoni is tall might be true and in another it might be false. This is in line with our intuitions.

YoniDanny

174 1940

Isaac

182

Gidi

176

Eli

196184.4

Average

Compositional Analysis (Positive Form)

• Consider the sentence: Yoni is tall• We make two assumptions:

• A covert POS (for positive) morpheme replaces the comparative morpheme –er and assigns truth conditions based on the degree of height that the individual has.

• The standard is determined by a context-dependent function stdctx that takes a gradable adjective as an input and returns the standard degree as an output.

Contextual Variability in Truth Conditions

• Consider the adjectives tall and expensive in:1. John is tall

2. The coffee in Rome is expensive

• Each sentence might be true or false, context-dependently: • (1) might be judged true if asserted as part of conversation about

John’s classmates, but false in a discussion on basketball players.

• (2) might be judged true in a discussion on the cost of living in various Italian cities, but false as part of a conversation about the cost of living in Chicago vs. Rome.

0

studentsbasketball players housesplants

john

Gradable Adjectives as Modifiers• Consider the sentences:

1. John is tall

2. John is a tall student

• According to the intersective analysis of modification, tall in (2) denotes a function of type (et)et:

[[tall]] = TALL-M = λfet. λxe.tall(x) ˄ f(x)

• It follows that:• (1) denotes: tall(john)• (2) denotes: tall(john) ˄ student(john)

• However, calculating the truth-value of tall(john) in (2) is done irrespectively of John being a student.

• Therefore, the context-variability that tall(john) introduces to (1) is introduced also to (2), without being restricted to students only.

• Our assumption that John is a tall student is equivalent to John is tall and John is a student is too simplistic.

• Note: for simplicity, we ignore degrees in this discussion of mod.

Gradable Adjectives as Modifiers – A Solution• The problem with the intersective analysis is that it ignores the

restrictiveness of tall: if John is a tall student , he is counted as tall among students (and he is a student by himself).

• Intuitively, we want tall to be restricted to students rather then being unrestricted.

• Define the attributive tall as a function of type (et)et as follows:[[tall]] = TALL-M = λfet. λxe.(TALL-R(λye.f(y)))(x) ˄ f(x)

• TALL-R is a function that takes a function – f – of type et as an argument and returns a function that takes an entity – x – and returns whether it is counted as tall among the entities that satisfy f, and it satisfies f.

• Conclusion: [[John is a tall student]] = TALL-R(λye.student(y))(john) ˄ student(john)

Properties of Relative Adjectives• Following Kennedy 2007 – Vagueness and Grammar.

• Under the state-of-the-art theory of gradable adjectives, the set of gradable adjectives is divided into two classes: Relative Adjectives and Absolute Adjectives. • Relative Adjectives: tall, expensive, large• Absolute Adjectives: clean, wet, full

• Properties of Relative Adjectives• The standard employed in their interpretation is context-variable.• They create borderline cases• They trigger the Sorites Paradox.

Borderline Cases• Say that the guy on the left is consider as tall while the

guy on the right isn’t.

90cm

30cm

Borderline Cases• What would you say about this guy?

62cm

Borderline Cases• What would you say about this guy?

• For any context, in addition to the sets of objects that a predicate like tall is clearly true of and clearly false of, there is typically a third set of objects for which it is difficult or impossible to make these judgments.

62cm

The Sorites Paradox• Consider the following premises:

• P1 - A $5 cup of coffee is expensive (for a cup of coffee).• P2 - Any cup of coffee that costs 1 cent less than an expensive one

is expensive (for a cup of coffee).

• The paradoxical conclusion from these rather reasonable premises is that even a 1$ cup of coffee is expensive. In fact, even a free cup of coffee would still be counted as expensive.

The Sorites Paradox• Premises:

• P1 - A theater with 1000 seats is big.• P2 - Any theater with 1 fewer seat than a big theater is big.

• Conclusion: any theater with 10 seats is big.

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FSP 2012 – GRADABILITY. Concept Properties like height, cleanliness and fullness are called Gradable since they may manifest to different extents over