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Quantification and Quantificational Structures

Overview 1. Interpreting DPs: entity, first-order quantification, generalized quantification 2. The possibility of interactions, with each other and 3. Expressivity of natural language quantification 4. Linguistically significant typology of quantifiers 5. Diversity at the interface: nominal and adverbial quantification

First Order Quantification

Represented by universal and existential operators that target entity-level variables.

With negation, the operators are inter-definable:

"x P(x) = ¬$x ¬P(x) ¬"x P(x) = $x ¬P(x) In illustrating logical forms for quantification below, we will use the following sentences:

1. Every dog barked 2. A dog barked 3. No dog barked 4. Fido chased every cat 5. Some dog chased Puff

In unrestricted variable format, the sentences have the following translations:

1. "x[dog¢(x) ® bark¢(x)] 2. $x[dog¢(x) Ù bark¢(x)] 3. ¬$x[dog¢(x) Ù bark¢(x)] 4. "x[cat¢(x) ® chase¢(f,x)] 5. $x[dog¢(x) Ù chase¢(x,p)]

In this format, the universal is always associated with the conditional connective and the existential with a conjunct. The truth of these formulas is determined only by facts concerning the individuals that satisfy (make true) the left-hand side of the formula—in the case of (1)-(3) and 5, facts about dogs. In the case of (4), facts about cats. Because of this general fact, the truth conditions expressed by these formulas can be expressed by representations in a formal language that allows the variables bound by the operators to be RESTRICTED by a . When using this notation, parentheses are generally put around the operator, and the variable. The following are translations using restricted variable quantification.

1. ("x:dog¢(x)) [bark¢(x)] 2. ($x:dog¢(x)) [bark¢(x)] 3. ¬($x:dog¢(x))[ bark¢(x)]

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4. ("x:cat¢(x))[chase¢(f,x)] 5. ($x:dog¢(x))[chase¢(x,p)]

Note that the structure of these representations more closely aligns with the structure of the English sentences that they translate. The contents of the Phrases (DP) provides the force and restriction of the binding operator. The SCOPE of the quantifier is provided by the remainder of the sentence. The force is expressed by the , which specify a template for the translation

1. every, allà ("v:RESTRICTION)[SCOPE] 2. a, some à ($v:RESTRICTION)[SCOPE] 3. no à ¬($v:RESTRICTION)[SCOPE]

Indeterminacy of Scope

Based upon the idea of a template translation, we can define a step-by-step process for building logical translations for more complex sentences, such as:

Every dog chased some cat

I. Divide the sentence by removing a quantificational DP and replacing it with a new variable. Repeat as needed until all quantificational DPs have been removed. II. Create a translation template for each DP by using the material in the NP complement of D to determine a restriction on the quantifier. III. Reassemble the templates into a single formula by using the formulas available to replace the scopes of the quantifiers.

Example:

I. Every dog:x + x chased some cat; Some cat:y + Every dog:x + x chased y II. ($y: cat¢(y))[SCOPE] + ("x: dog¢(x))[SCOPE] + chase¢(x,y) III. ($y: cat¢(y))[SCOPE] + ("x: dog¢(x))[ chase¢(x,y)] ($y: cat¢(y))[("x: dog(x)¢)[ chase¢(x,y)]]

Note that in sentences with multiple quantificational DPs, there can be alternative results, depending upon the order of reassembly.

Quantification as Higher Order Predication

First order formulas like these can capture the truth conditions of many, but not all, of natural language quantification. For example, the truth condition expressed by most cannot be given a first order translation.

Generalized quantifier theory treats quantificational determiners as expressing second order predication: relations between the sets that are the of the restriction and scope predicates. The sentence in (1) can be given second order representations as below:

1. every¢ (dog¢, bark¢) = every¢ (lx dog¢(x)) (ly bark¢(y))

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The definition of the determiners:

a) every¢ (A, B) = 1 iff AÍ B. b) some¢ (A, B) = 1 iff AÇB ≠ Æ c) no¢ (A, B) = 1 iff AÇB = Æ

Within this framework, other determiners can be defined with conditions on the size of the intersections of their restriction and scope arguments. Other determiners can be defined in terms of comparison of the size of those intersections to the size of their restrictions. The former are called intersectional determiners and are symmetric relations. The latter are called proportional determiners, and are not symmetric relations. Some determiners have both an intersectional and a proportional interpretation.

The interpretation of quantifiers in are subject to various types of parameterization, most notably contextual relevance of the individuals in the restriction (the base of the quantifier), and standards of comparison or expectation for scalar ordered values such as a few, several, many, a lot.

Linguistically Significant Classes of Quantifiers

Intersective/symmetric Proportional/asymmetric

Strong vs. weak DPs and the effect Milsark’s generalization: Properties can only be predicated of strong DPs. Existential pivots must be weak DPs.

Monotonicity of quantifiers:

The quantificational force of a DP determines whether it is monotonically INCREASING or DECREASING with respect to its scope.

For increasing quantifiers: For B Í C, then D(A,B) entails D(A,C) For decreasing quantifiers: For B Í C, then D(A,C) entails D(A,B)

Walk slowly Í walk: Every man walks slowly implies Every man walks. No man walks implies No man walks slowly.

Adverbial Quantification

The presentation above exploits a structural parallel between quantification expressed in DP structures and the semantic notions of restriction and scope. Quantification is not always expressed in such structures; it can also be adverbial.

1. Fido usually chases a cat. 2. Puff sometimes eats sardines. 3. Cats rarely chase dogs.

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In such cases, interpretation is based upon fixing the type of quantification (entity vs. eventuality) and determining the intended restriction and scope in context. and background construal constrains these.

In some languages, adverbial quantification is the primary if not the only structural expression of these relations.

As background reading: the selections from the Kearns text and from the Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/logical-form/ https://plato.stanford.edu/entries/generalized-quantifiers/