SoSE15: Term paper project: Determiners

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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 determiners in the following sentences.

(a) Olaf talks to some reindeer.

(b) Every human sings.

(c) The queen has a sibling.

Check your solutions here:

(a) some

(b) every

(c) the, a

Task 2: Identify the correct formula for this paraphrase.

Every reindeer which eats carrots sings.

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

Participants

• Monique Lanz

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