Semantics1 Week 2

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Additional material for the meeting of week 2, April 22, 2015.

Our literary scenario: Game of Thrones (TV series)

Our scenario and some preliminary thoughts

Getting into our literary scenario

Watch the trailer of Season 1 of Game of Thrones:


Having watched the video, which of the following statements are true in our scenario?

Click on those statements that are true in our scenario.

John Snow is Ned Stark's son.
Catelyn wants that her husband becomes the king's Hand.
Catelyn wants her husband to become the Hand of the king.
There is a king.
There is no king.


The meanings of some of these sentences are closely related. What do you observe for the following sentence pairs?

Catelyn wants that her husband becomes the king's Hand. and Catelyn wants her husband to become the Hand of the king.

Check your answers

The two sentences are paraphrases of each other, i.e., in every situation, whever the first is true, so is the second.

There is a king. and There is no king.

Check your answers

The two sentences contradict each other. Whenever one is true, the other must be false.


Catelyn wants that her husband becomes the king's Hand. and There is a king.

Check your answers

Whoever utters the first sentence must also assume the truth of the second. (Technically, the second sentence is presupposed by the first, but this doesn't matter here.)

Task:

  1. Formulate three more statements with respect to our scenario.
  2. Determine for each of them whether it is true or false in our scenario.
  3. Is there a systematic relation between the meaning of your sentences?

Why it is too difficult to go directly from language to the world

The following architecture is extremely useful when talking about semantics:

  1. A natural language expressions: Rob likes John.
  2. ... is mapped to some expression from a formal language (here: predicate logic): like(rob,john)
  3. This logical expression is then interpreted with respect to our scenario/world: The formula like(rob,john) is true, because, in our scenario, Rob likes John.


The following properties of natural language make it useful to use the intermediate step of a logical language:

  1. The same expression can have different meanings (ambiguity).
  2. Different expressions can have the same meaning (synonyms, paraphrases)

Find examples for the above-mentioned properties (ambiguity, synonymy, paraphrases).

Check your answers

1. one form, two meaingns: Ambiguity: (see earlier in this meeting and the slides of last week's meeting)

1.a Ambiguous words: date (fruit or point in time); bank (financial institute or bank of a river)

1.b. Ambiguous sentences: Ned

2. two forms, one meaning:

2.a Synonymous words: couch - sofa; instant - moment

2.b Paraphrases:

  • active-passive pairs: Robert invited Ned to King's Landing. - Ned got invited to King's Landing by Robert.
  • cleft sentences: Robert invited Ned to King's Landing. - It was to King's Landing that Robert invited Ned.
  • our previous example: Catelyn wants that her husband becomes the king's Hand. and Catelyn wants her husband to become the Hand of the king.


Towards a formal model

First steps

You can think of building a formal model like being the producer of a film who has to collect everything that should be included in the film.

Here is a very simple story from which we can derive an example model.

Mark those elements that we need in a model.

relations
individuals
nouns
properties
relatives


What is the status of the following entities in the video on Little Red Riding Hood?

individualpropertyrelation
Red Riding Hood
lives in the forest
Grandmother
is afternoon snack for
has a red hood
has a big mouth
is grandmother of


The universe and name symbols

Task: Assume three individuals from our Game of Thrones-scenario.

Formally we collect the individuals of our model in a so-called universe (U). For the fairy-tale story, we can define the universe as follows:

U = {Redridinghood, Grandmother, Wolf}

Do a similar definition for your own scenario.


We can introduce name symbols for some of our individuals. For example: redridinghood, grandmother, wolf.

We link the name symbols to the individuals in our modal. To do this, we introduce the interpretation function. We will written the interpretation function as as I.
This function can be defined in the following way:

I(grandmother) = Grandmother
I(redridinghood) = Red Riding Hood
I(wolf) = Wolf

Relations and predicate symbols

In the fairy-tale scenario we express a relation between Little Red Riding Hood and the Wolf, namely that Little Red Riding Hood is the Wolf's afternoon snack. To formalize this, we collect all pairs of individuals which are such that the first element in the pair is the afternoon snack of the second. Note: A pair is written in between pointy brackets.


Formally we can write this down as follows:
{< x, y > | x is y 's afternoon snack} = { < Redridinghood, Wolf >, < Grandmother, Redriding hood >.}

We can also assume empty relations:

{< x, y > | x is y 's father } = { }


Note, if a relation works both ways, two pairs must be added:

{< x, y > | x talks with y} = { <Redridinghood, Wolf >, < Wolf, Redridinghood >}


Task: Using your Game of Thrones-universe from above, introduce one binary and one ternary relation.

Just like with names, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols afternoon-snack-of_2 and father-of_2, and talks-with_2. (The number 2 indicates that the interpretation consists of pairs, not just of single individual) There interpretation is defined as follows:

I(afternoon-snack-of_2) = { < x, y > | x is y 's afternoon snack } = { <Redridinghood, Wolf >, <Grandmother, Wolf > }.

Task: For each of your properties, invent an appropriate predicate symbol. Define its interpretation.

Properties and predicate symbols

A property is a specification that either holds of an individual or not. In the little story, having a big mouth is a property of the Wolf, but of noone else in the story. Being female holds of both Little Red Riding Hood and the Grandmother.

We can think of a property as the set of individuals that have this property. Under this view, the property of being female would be the set {Redridinghood, Grandmother}.

Alternatively it is convenient to think of properties as 1-place relations. Under this view, the property of being female would be a set of lists of length 1. This is what the property of being female then looks like: { <Redridinghood>, <Grandmother> }

Task: Using your Game of Thrones universe, define two properties in the format of 1-place relations.

Just like before, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols female_1 and has-big-mouth_1. There interpretation is defined as follows:

I(female_1) = { < x > | x is female } = { <Redridinghood>, <Grandmother> }.

Task: For each of your properties, invent an appropriate predicate symbol. Define its interpretation.

For next week


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