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 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. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

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 number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

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. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

Model from the scenario "Frozen"

Inspired by the Disney movie "Frozen" released in 2013.

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>}

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 of 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	✗	✓
Sven	✗	✗
Olaf	✗	✗
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.

Exercises

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

Task 1: Identify the correct formula for these paraphrases.

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. Explain further, if and why the formulae are true or false.

1. Some reindeer is male and royal.
2. The snowman sings.
3. Every queen likes Anna.

Check your solutions here:

1. ∃x (reindeer₁(x) : (male₁(x) ∧ royal₁(x))) → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
2. sing₁(ιx : snowman₁(x)) → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
3. ∀x (queen₁(x) : like₂(x, Anna)) → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.

Task 3: Write down the correct sentences for the logical formulae in the boxes.

1  

∃x (prince1(x) : love2(Elsa, x))

2  

∀x ((woman1(x) ∧ red-haired1(x)) : dance-with2(x, Hans))

3  

like2((ιx : iceman1(x)), Sven)

References

Kearns, Kate. 2000. Semantics. 42-47, 68-119. New York: Palgrave Macmillan.

Levine, Robert D., Frank Richter & Manfred Sailer. 2015. Formal Semantics. An Empirically Grounded Approach. 203-248. Stanford: CSLI Publications.

Participants

• Monique Lanz

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