SoSE15: Term paper project: Determiners: Difference between revisions
Monique Lanz (talk | contribs) |
Monique Lanz (talk | contribs) |
||
Line 8: | Line 8: | ||
* Three exercises for each operator | * Three exercises for each operator | ||
= The difference between | = The difference between logical quantifiers and definite descriptions = | ||
The universal and existential quantifiers have to be interpreted differently than the definite article. | The universal and existential quantifiers have to be interpreted differently than the definite article. |
Revision as of 17:41, 21 August 2015
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 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 individual 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 has rather to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.
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>}
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 | ✗ | ✓ |
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.
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.
Task 2: Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.
- Some reindeer is male and royal.
- The snowman sings.
- Every queen likes Anna.
Check your solutions here:
- ∃x (reindeer1(x) : (male1(x) ∧ royal1(x))) → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
- sing1(ιx : snowman1(x)) → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
- ∀x (queen1(x) : like2(x, Anna)) → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.
Task 3: Write down the correct sentences for the logical formulae in the boxes.
References
Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.
Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.
Participants
Back to the Semantics 2 page.