Sentences described in Lambda terms

[kimmmmi]

Homework Statement

In a sentence like "Tina is tall", "Tina" is an entity (type e), "is" is an identity function and "tall" is an function of entities to truth values (e-> t or et). The denotations are something like this: Tina = Tina' , is = $$\lambda$$x.x and tall = tall'. So in lambda terms the whole sentence is something like (thin'(Tina')). But my problems are:
1. I don't see any lambda terms in this description
2. If the sentence gets more complicated with logical functions like AND ($$\wedge$$) and OR ($$\vee$$), I'm not sure what the overall lambda term should look like.
3. More specificially: are there some kind of rules to follow (a "stappenplan" in dutch) to get from denotations to lambda terms for the entire sentence?2. An example

Tina is tall or (Tina is thin and Tina is not thin).
We know that this sentence is equivalent to Tina is tall, because Tina cannot be thin and not thin at the same time, so the right constituent of the OR is false. How can we give a lambda term for this sentence?

The Attempt at a Solution

WORD TYPE DENOTATION
Tina e Tina'
is (et)(et) $$\lambda$$x.x
tall et tall'
or t(tt) $$\lambda$$x.$$\lambda$$y.$$\vee$$t(tt) (x)(y)
thin et thin'
and (et)(et)(et) $$\lambda$$x.$$\lambda$$y.$$\lambda$$z.$$\wedge$$et((et)(et)) x(z)y(z)
not (et)(et) ?? $$\lambda$$x.-x

The lambda term has to be something like this right?:

$$\vee$$(tall'(Tina')) ($$\wedge$$(thin'(Tina')) (-(thin'(Tina')))
But where did all the lambda terms go??

 

[mighty2000]

1. The lambda terms are not explicitly written out in the description, but they are implied by the use of the lambda calculus notation. For example, "is" is represented by the lambda abstraction \lambdax.x, where x represents the entity being described. Similarly, "tall" is represented by the lambda abstraction tall', where tall' is a function of entities to truth values.

2. The overall lambda term for a sentence with logical functions such as AND and OR will depend on the specific sentence and its logical structure. In general, you can use the rules of lambda calculus to build the lambda term step by step. For example, the lambda term for "Tina is tall or (Tina is thin and Tina is not thin)" could be built as follows:

a. Start with the denotations for the individual words: Tina' for "Tina", \lambdax.x for "is", tall' for "tall", thin' for "thin", and \lambdax.-x for "not".
b. Use the lambda abstraction \lambdax.\lambday.\veet(tt) to represent the logical OR function, where t represents a truth value and the two x's represent the two propositions being connected by the OR.
c. Use the lambda abstraction \lambdax.\lambday.\lambdaz.\wedgeet((et)(et)) to represent the logical AND function, where the two x's represent the two propositions being connected by the AND and z represents a truth value.
d. Substitute the denotations for each word into the appropriate lambda abstraction, taking into account the type restrictions for each function. For example, for the word "not", the denotation is \lambdax.-x, but it cannot be directly substituted into the OR function, which expects two propositions of type (et)(et). So we need to first apply the denotation for "not" to the denotation for "thin", resulting in \lambdax.-thin'(x), which can then be substituted into the OR function.
e. Continue substituting and simplifying until the final lambda term is reached. In this case, the final lambda term would be: \vee(tall'(Tina')) (\wedge(thin'(Tina')) (-thin'(Tina'))).

3. The steps outlined above can be seen as a "stappenplan" for