Semantics 1, WiSe 2016/17, Week 3

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: Miranda loves Ferdinand.
2. ... is mapped to some expression from a formal language (here: predicate logic): love2(miranda,ferdinand)
3. This logical expression is then interpreted with respect to our scenario/world: The formula love2(miranda,ferdinand) is true, because, in our scenario, Miranda loves Ferdinand.

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: Sycorax and Prospero were stranded on the island with their children.

2. two forms, one meaning:

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

2.b Paraphrases:

• active-passive pairs: Prospero set Ariel free. - Ariel was set free by Prospero.
• cleft sentences: Prospero set Ariel free. - It was Prospero who set Ariel free.
• different ways to express a possessor: Sycorax was the first inhabitant of the island. and Sycorax was the island's first inhabitant.

Towards a formal model

First steps

The following material is an adapted form of material created by student participants of the project e-Learning Resources for Semantics (e-LRS). Involved participants: Lisa, Marthe, Elisabeth, Isabelle.

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?

individual	property	relation
			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 >}

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.

Computing the truth value of atomic formulae

The following video presents the step-by-step computation of the truth value of two atomic formulae. The example uses a model based on Shakespeare's play Macbeth. The two formulae are:

• kill(macbeth,duncan)
• kill(lady-macbeth,macbet)

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