Logicality and Semantic Theory Gil Sagi University of Haifa July - - PowerPoint PPT Presentation

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Logicality and Semantic Theory Gil Sagi University of Haifa July 26, 2018 Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 1 / 36 Introduction The Concept of Logical Consequence Gil Sagi (Haifa University)

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Logicality and Semantic Theory

Gil Sagi

University of Haifa

July 26, 2018

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 1 / 36

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Introduction

The Concept of Logical Consequence

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 2 / 36

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Introduction

The Concept of Logical Consequence

All Greeks are human All humans are mortal All Greeks are mortal

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 2 / 36

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Introduction

The Concept of Logical Consequence

All Greeks are human All humans are mortal All Greeks are mortal

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 2 / 36

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Introduction

Is there logic in natural language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic? What is meant by Natural Language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic? What is meant by Natural Language? What is meant by Logic in Natural Language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic? Criteria for logicality: Invariance under isomorphisms What is meant by Natural Language? What is meant by Logic in Natural Language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic? Criteria for logicality: Invariance under isomorphisms What is meant by Natural Language? An empirical phenomenon, the subject mater of current semantic theory What is meant by Logic in Natural Language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Is there logic in natural language?

What is meant by Logic? Criteria for logicality: Invariance under isomorphisms What is meant by Natural Language? An empirical phenomenon, the subject mater of current semantic theory What is meant by Logic in Natural Language? Is there a relation satisfying the criterion of invariance under isomorphisms that is modeled by current semantic theory?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 3 / 36

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Introduction

Logic as Model

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 4 / 36

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Introduction

Logic as Model

A formal language with its syntax and semantics can be used to model logical consequence in natural language. [Shapiro, 1998]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 4 / 36

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Introduction

Outline

1

Introduction

2

Glanzberg A: “Logical Consequence in Natural Langauge” (2015)

3

Semantic Constraints The Framework Invariance Criteria

4

Glanzberg B: “Explanation and Partiality in Semantic Theory” (2014)

5

Conclusion

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 5 / 36

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Glanzberg A

Glanzberg

[T]he logic in natural language thesis: A natural language, as a structure with a syntax and a semantics, thereby determines a logical consequence relation. (Glanzberg, 2015)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 6 / 36

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Glanzberg A

Glanzberg

[T]he logic in natural language thesis: A natural language, as a structure with a syntax and a semantics, thereby determines a logical consequence relation. (Glanzberg, 2015) Glanzberg: The logic in natural language thesis is false (assuming a restrictive notion of logic).

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 6 / 36

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Glanzberg A

The Argument from Absolute Semantics

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes

[Chierchia and McConnell-Ginet 2000; Heim and Kratzer 1998]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 7 / 36

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Glanzberg A

The Argument from Lexical Entailment

  • a. We loaded the truck with hay.

ENTAILS We loaded hay on the truck.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 8 / 36

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Glanzberg A

The Argument from Lexical Entailment

  • a. We loaded the truck with hay.

ENTAILS We loaded hay on the truck.

  • b. We loaded hay on the truck.

DOES NOT ENTAIL We loaded the truck with hay.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 8 / 36

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Glanzberg A

The Argument from Lexical Entailment

John cut the bread. ENTAILS The bread was cut with an instrument.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 9 / 36

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Glanzberg A

The Argument from Logical Constants

most(A, B) ⇔ |A\B| < |A ∩ B|

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 10 / 36

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Glanzberg A

The Argument from Logical Constants

  • a. Local: mostM = {A, B ∈ M2 : |A\B| < |A ∩ B|}
  • b. Global: function from M to mostM

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 11 / 36

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Glanzberg A

Glanzberg

[W]e only get to logic proper by a significant process of identification, abstraction, and idealization. We first have to identify what in a language we will count as logical constants. Afer we do, we still need to abstract away from the meanings of non-logical expressions, and idealize way from a great many features of languages to isolate a consequence relation. This process takes us well beyond what we find in a natural language and its semantics. We can study logic by thinking about natural language, but this sort

  • f process shows that we will need some substantial extra-linguistic

guidance to—some substantial idea of what we think logic is supposed to be—to do so. We do not get logic from natural language all by itself. (Glanzberg, 2015)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 12 / 36

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Glanzberg A

Invariance under Isomorphisms and Semantic Theory

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 13 / 36

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Glanzberg A

Invariance under Isomorphisms and Semantic Theory

Invariance under isomorphisms as a criterion for logicality: is this mathematical property representative of a linguistic distinction?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 13 / 36

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Glanzberg A

Invariance under Isomorphisms and Semantic Theory

Logical terms in natural language

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 14 / 36

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Glanzberg A

Invariance under Isomorphisms and Semantic Theory

Logical terms in natural language There aren’t: [Harman, 1984, Lycan, 1984]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 14 / 36

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Glanzberg A

Invariance under Isomorphisms and Semantic Theory

Logical terms in natural language There aren’t: [Harman, 1984, Lycan, 1984] There are: [Fox, 2000, Gajewski, 2002, Fox and Hackl, 2006]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 14 / 36

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Semantic Constraints The Framework

Semantic Constraints

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 15 / 36

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Semantic Constraints The Framework

Semantic Constraints

Fixing something amounts to limiting the admissible interpretations.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 15 / 36

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Semantic Constraints The Framework

Semantic Constraints

Fixing something amounts to limiting the admissible interpretations. (∧): I(ϕ ∧ ψ) = T ⇔ I(ϕ) = T and I(ψ) = T

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 15 / 36

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Semantic Constraints The Framework

Semantic Constraints

Fixing something amounts to limiting the admissible interpretations. allGreen allRed,

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 16 / 36

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Semantic Constraints The Framework

Semantic Constraints

Fixing something amounts to limiting the admissible interpretations. I(allGreen) I(allRed),

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 17 / 36

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Semantic Constraints The Framework

Semantic Constraints

Fixing something amounts to limiting the admissible interpretations. I(allRed) ∩ I(allGreen) = ∅

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 18 / 36

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Semantic Constraints The Framework

I(Ann) = Ann

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D} I(R) is a symmetric binary relation.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D} I(R) is a symmetric binary relation. 0 ∈ I(naturalNumber)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D} I(R) is a symmetric binary relation. 0 ∈ I(naturalNumber) I(prime) = {2, 3, 5, ...}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(Ann) = Ann I(smokes) = λx ∈ D. x smokes I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|} I(even) ∩ I(odd) = ∅ I(bachelor) ⊆ I(unmarried) I(H2O) = I(water) I(wasBought) = I(wasSold) I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D} I(R) is a symmetric binary relation. 0 ∈ I(naturalNumber) I(prime) = {2, 3, 5, ...} |I(Red)| = 375 (i.e., the size of the extension of Red is 375.)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

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Semantic Constraints The Framework

I(P) ⊆ D

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

I(P) ⊆ D I(John) ∈ D

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

I(P) ⊆ D I(John) ∈ D I(abc) = T or I(abc) = F

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

I(P) ⊆ D I(John) ∈ D I(abc) = T or I(abc) = F I(s) = T

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

I(P) ⊆ D I(John) ∈ D I(abc) = T or I(abc) = F I(s) = T I(d) = I(∧)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

I(P) ⊆ D I(John) ∈ D I(abc) = T or I(abc) = F I(s) = T I(d) = I(∧) I(or) ∈ {f∨, f⊻} where f∨ is the inclusive or function, and f⊻ is the xor function from pairs of truth values to truth values.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

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Semantic Constraints The Framework

The Language and its Models

Language

Primitive expressions (terms) Complex expressions (phrases)

Models

M = D, I D (the domain) is a non-empty set. I (the interpretation function) assigns to phrases values from the set-theoretic hierarchy with the members of D ∪ {T, F} as ur-elements.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 21 / 36

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Semantic Constraints The Framework

Semantic Constraints

A semantic constraint for L is a sentence in the metalanguage that somehow constrains or limits the admissible models for L. Semantic constraints include implicit universal quantification over models (domains and interpretation functions).

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 22 / 36

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Semantic Constraints The Framework

Semantic Constraints

A semantic constraint for L is a sentence in the metalanguage that somehow constrains or limits the admissible models for L. Semantic constraints include implicit universal quantification over models (domains and interpretation functions). Let ∆ be a set of semantic constraints. A ∆-model is an admissible model by ∆, i.e. a model abiding by the constraints in ∆.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 22 / 36

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Semantic Constraints The Framework

Logical Consequence

Let ∆ be a set of constraints such as those mentioned above. An argument Γ, ϕ is ∆-valid (Γ | =∆ ϕ) if for every ∆-model M, if all the sentences in Γ are true in M, then ϕ is true in M. So, for instance we have: bachelor(John) | =∆ unmarried(John).

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 23 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms:

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t, we have an associated

  • peration Ot such that for each set D, Ot(D) gives the extension of t in

models with domain D.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t, we have an associated

  • peration Ot such that for each set D, Ot(D) gives the extension of t in

models with domain D. Examples: O∃(D) = {A ⊆ D : A = ∅}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t, we have an associated

  • peration Ot such that for each set D, Ot(D) gives the extension of t in

models with domain D. Examples: O∃(D) = {A ⊆ D : A = ∅} O∀(D) = {D}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t, we have an associated

  • peration Ot such that for each set D, Ot(D) gives the extension of t in

models with domain D. Examples: O∃(D) = {A ⊆ D : A = ∅} O∀(D) = {D} O∃ℵ0(D) = {A ⊆ D : |A| ≥ ℵ0}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: Let M = D, I and M′ = D′, I′ be models, and let f : D → D′ be a bijection.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: Let M = D, I and M′ = D′, I′ be models, and let f : D → D′ be a bijection. f can be naturally extended to f + over elements in the set-theoretic hierarchy over D ∪ {T, F} and D′ ∪ {T, F}.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for logical terms: Let M = D, I and M′ = D′, I′ be models, and let f : D → D′ be a bijection. f can be naturally extended to f + over elements in the set-theoretic hierarchy over D ∪ {T, F} and D′ ∪ {T, F}. Definition (invariance under isomorphisms: terms) A term t is invariant under isomorphisms if for any sets D and D′ and a bijection f : D → D′, f +(Ot(D)) = Ot(D′).

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for semantic constraints:

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for semantic constraints: Definition (isomorphic models) We say that M = D, I is isomorphic to M′ = D′, I′ (M ∼ = M′) if there is a bijection f : D → D′ such that f +(I(p)) = I′(p) for every phrase p in L.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism As a criterion for semantic constraints: Definition (isomorphic models) We say that M = D, I is isomorphic to M′ = D′, I′ (M ∼ = M′) if there is a bijection f : D → D′ such that f +(I(p)) = I′(p) for every phrase p in L. Definition (invariance under isomorphism: semantic constraints) A semantic constraint C is invariant under isomorphisms if for any models M and M′ such that M ∼ = M′, then if M is a {C}-model, then M′ is a {C}-model. (cf. [Zimmermann, 2011])

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

The following constraints are invariant under isomorphisms:

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

The following constraints are invariant under isomorphisms: I(allRed) ∩ I(allGreen) = ∅ I(Red) ∩ I(Big) = ∅ |I(Red)| = 375 (i.e., the size of the extension of Red is 375.) I(John) ∈ I(Bachelor) I(s) = T I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

The following constraints are invariant under isomorphisms: I(allRed) ∩ I(allGreen) = ∅ I(Red) ∩ I(Big) = ∅ |I(Red)| = 375 (i.e., the size of the extension of Red is 375.) I(John) ∈ I(Bachelor) I(s) = T I(∃) = {A ⊆ D : A = ∅} I(∀) ∈ {{B ⊆ D : A ⊆ B} : A ⊆ D} The following constraints are not invariant under isomorphisms: 0 ∈ I(naturalNumber) I(prime) = {2, 3, 5, ...} I(Even) ∩ I(Prime) = {2} I(Ann) = Ann I(smokes) = λx ∈ D. x smokes

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism: terms and constraints

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism: terms and constraints Consider the constraint Ct that “fixes” t: Ct : I(t) = Ot(D)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism: terms and constraints Consider the constraint Ct that “fixes” t: Ct : I(t) = Ot(D) Example: C= : I(=) = {a, a : a ∈ D}

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

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Semantic Constraints Invariance Criteria

Criteria for Semantic Constraints

Invariance under isomorphism: terms and constraints Consider the constraint Ct that “fixes” t: Ct : I(t) = Ot(D) Example: C= : I(=) = {a, a : a ∈ D}

  • Proposition. Let t be a term, Ot an associated operation and Ct an

associated constraint. Then t is invariant under isomorphisms iff Ct is invariant under isomorphisms.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

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Semantic Constraints Invariance Criteria

Can we apply the generalized criterion of invariance under isomorphisms to natural language semantics, and in this way demarcate the relation of logical consequence in natural language?

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 29 / 36

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Glanzberg B

Partiality in Semantic Theory

[S]emantics, narrowly construed as part of our linguistic competence, is only a partial determinant of content. Likewise, semantic theories in linguistics function as partial theories of

  • content. I shall go on to offer an account of where and how this

partiality arises, which focuses on how lexical meaning combines elements of distinctively linguistic competence with elements from

  • ur broader cognitive resources. This account shows how we can

accommodate some partiality in semantic theories without falling into skepticism about semantics or its place in linguistic theory. (Glanzberg, 2014)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 30 / 36

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SLIDE 79

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann

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SLIDE 80

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes

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SLIDE 81

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes
  • c. most(A, B) ⇔ |A\B| < |A ∩ B|

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

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SLIDE 82

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes
  • c. most(A, B) ⇔ |A\B| < |A ∩ B|
  • d. tall(x) = d ∈ S (S a scale with the dimension of height)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

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SLIDE 83

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes
  • c. most(A, B) ⇔ |A\B| < |A ∩ B|
  • d. tall(x) = d ∈ S (S a scale with the dimension of height)
  • e. short(x) = d ∈ S′ (S′ the inverse of S)

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SLIDE 84

Glanzberg B

Uninformative Semantic Clauses

  • a. Ann = Ann
  • b. smokes = λx ∈ De : x smokes
  • c. most(A, B) ⇔ |A\B| < |A ∩ B|
  • d. tall(x) = d ∈ S (S a scale with the dimension of height)
  • e. short(x) = d ∈ S′ (S′ the inverse of S)

[T]he use of disquotation in semantic theories precisely marks the places where they lose their explanatory force. Insofar as disquotation plays an ineliminable role in building theories of content, semantic theories can be at best partial theories of content... [D]isquotation is a guide to where linguistic meaning contains pointers to extra-linguistic elements of content. (Glanzberg, 2014)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

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SLIDE 85

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann

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SLIDE 86

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D

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SLIDE 87

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes

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SLIDE 88

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes I(smokes) ∈ {f : f : D → {T, F}}

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SLIDE 89

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes I(smokes) ∈ {f : f : D → {T, F}} I(tall)(x) ∈ S (S a scale with the dimension of height)

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

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SLIDE 90

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes I(smokes) ∈ {f : f : D → {T, F}} I(tall)(x) ∈ S (S a scale with the dimension of height) I(tall)(x) ∈ S (S a scale on the vertical axis....)

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SLIDE 91

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes I(smokes) ∈ {f : f : D → {T, F}} I(tall)(x) ∈ S (S a scale with the dimension of height) I(tall)(x) ∈ S (S a scale on the vertical axis....) I(short)(x) ∈ S (S′ the inverse of S)

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SLIDE 92

Glanzberg B

Separating the Explanatory from the Non-Explanatory

I(Ann) = Ann I(Ann) ∈ D I(smokes) = λx ∈ D. x smokes I(smokes) ∈ {f : f : D → {T, F}} I(tall)(x) ∈ S (S a scale with the dimension of height) I(tall)(x) ∈ S (S a scale on the vertical axis....) I(short)(x) ∈ S (S′ the inverse of S) I(most) = {A, B ∈ P(D)2 : |A ∩ B| > |A\B|}

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SLIDE 93

Glanzberg B

Logical Consequence in Natural Language

The generalized criterion of invariance under isomorphisms: A semantic constraint is logical if it is invariant under isomorphisms.

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 33 / 36

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SLIDE 94

Glanzberg B

Logical Consequence in Natural Language

The generalized criterion of invariance under isomorphisms: A semantic constraint is logical if it is invariant under isomorphisms. Conjecture: The logic of natural language is precisely the explanatory part of semantic theory for natural language.

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SLIDE 95

Conclusion

Refined Criterion for Logicality

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SLIDE 96

Conclusion

Refined Criterion for Logicality

A connective is a logical connective if and only if it follows from the meaning of the connective that it is invariant under arbitrary

  • bijections. [McGee, 1996, p. 578]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 34 / 36

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SLIDE 97

Conclusion

Refined Criterion for Logicality

A semantic constraint is a logical semantic constraint if and only if it follows from the meaning of the semantic constraint that it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 35 / 36

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SLIDE 98

Conclusion

Refined Criterion for Logicality

A semantic constraint is a logical semantic constraint if and only if it follows from the semantic theory for the language and it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578]

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 36 / 36

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SLIDE 99

Conclusion

Refined Criterion for Logicality

A semantic constraint is a logical semantic constraint if and only if it follows from the semantic theory for the language and it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578]

Thank you!

Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 36 / 36

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SLIDE 100

Conclusion

Fox, D. (2000). Economy and Semantic Interpretation, Linguistic Inquiry Monographs 35. MITWPL and MIT Press, Cambridge, MA. Fox, D. and Hackl, M. (2006). The universal density of measurment. Linguistics and Philosophy, 29:537–586. Gajewski, J. (2002). On analyticity in natural language. Manuscript. Harman, G. (1984). Logic and reasoning. Synthese, 60:107–127. Lycan, W. (1984). Logical Form in Natural Language. The MIT Press, Cambridge, MA.

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SLIDE 101

Conclusion

McGee, V. (1996). Logical operations. Journal of Philosophical Logic, 25:567–580. Shapiro, S. (1998). Logical consequence: Models and modality. In Schirn, M., editor, The Philosophy of Mathematics Today, pages 131–156. Oxford Univerity Press, Oxford. Zimmermann, T. E. (2011). Model-theoretic semantics. In Maienborn, C., von Heusinger, K., and Portner, P., editors, Semantics: an international handbook of natural language meaning, volume 33. Walter de Gruyter.

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