SoSE15: Term paper project: Determiners

Warning:
The material on this page has been created as part of a seminar. It is still heavily under construction and we do not guarantee its correctness. If you have comments on this page or suggestions for improvement, please contact Manfred Sailer.
This note will be removed once the page has been carefully checked and integrated into the main part of this wiki.

Short description of the project

• Difference between "every", "some" and the definite article;
• Video about how to differentiate "some" and the definite article;
• Three exercises for each operator

The difference between the logical quantifiers and definite descriptions

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that every single individual in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is at least one individual or more in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is absolutely one individual and no more or less in a model that fits exactly the described features.

Model from the scenario "Frozen"

Individuals:

• Elsa, the Snow Queen of Arendelle
• Anna, the Princess of Arendelle
• Kristoff, an iceman
• Sven, a reindeer
• Olaf, a snowman
• Hans, the Prince of the Southern Isles

Properties:

• royal₁ = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}
• prince₁ = {<x> | x is a prince} = {<Hans>}
• human₁ = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}
• male₁ = {<x> | x is male} = {<Kristoff>, <Sven>, <Olaf>, <Hans>}

Relations:

• sibling₂ = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}
• get-engaged₂ = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}

Example for the universal quantifier

Example: Every royal is human.

Here, every is our determiner, royal is our restrictor and human is the scope. We will choose x as our variable. Therefore, the paraphrase would look like that:

For every x such that x is royal x is human.

The overall formula for this expression would look like that:

∀ x (royal₁(x) : human₁(x))

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

g(x)	royal1(x)	human1(x)
Elsa	✓	✓
Anna	✓	✓
Kristoff	x	✓
Sven	x	x
Olaf	x	x
Hans	✓	✓

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

The Difference between the existential quantifier and the definite article

Here is a video about how to differentiate the existential quantifier and the definite article:

Exercises

After having read this page and watched the video, work on the following tasks:

Task 1: Identify the correct formula for this paraphrase.

1 Every reindeer which eats carrots sings.

	∀x (reindeer1(x) : (eat-carrots1(x) ∧ sing1(x)))
	∀x (reindeer1(x) ∧ eat-carrots1(x) ∧ sing1(x))
	∀x ((reindeer1(x) ∧ eat-carrots1(x)) : sing1(x))

2 A snowman loves summer.

	love2((∃x snowman1(x)), summer)
	∃x (snowman1(x) : love2(x, summer))
	∃x (snowman1(x) : love2(summer))

3 The iceman loves a princess.

	∃x (princess1(x) : love2(iceman, x)
	∃x (princess1(x) : love2((ιy : iceman1(y)), x))
	ιx : (iceman1(x) ∧ love2(x, ∃y (princess1(y))))

Task 2: Write down the logical formulae for the following sentences.

(a) Some reindeer is male and royal.

(b) The snowman sings.

(c) Every queen likes Anna.

Check your solutions here:

(a) ∃x (reindeer₁(x) : (male₁(x) ∧ royal₁(x)))

(b) sing₁(ιx : snowman₁(x))

(c) ∀x (queen₁(x) : like₂(x, Anna))

Participants

• Monique Lanz

---

Back to the Semantics 2 page.