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My exposure so far to type theory in formal semantics has included examples of rather simple phrases and sentences only. Individual constants of are type e (e.g. John, Chicago, etc.), intransitive verbs are of type <e, t> (e.g. sleep, laugh), transitive verbs are of type <e,<e,t>> (e.g. throw, grab), many adjectives are like intransitive verbs in that they are of type <e, t> (e.g. happy, red).

I am looking for a more extensive inventory of these types for English word categories. I would like to be able to play with the combinatory power of the system, but in order to do this, I need a much more extensive inventory than one typically encounters. In this regard, can every word category be assigned a type, e.g. auxiliary verbs (is, has, does), modal verbs (might, may, will, etc.)? Are there distinct types for the various forms of content verbs (finite form, infinitive, present participle, past participle)? What about the negation not; what is its type? What is the type of a purely functional preposition such as of in the picture of John)?

As I understand formal semantics so far, every word category should indeed be assigned a type in order for the system to be truly compositional and to hence match the compositional nature of syntactic structures. I think that in this regard, every single word should be assigned a type, which means someone must have created an extensive inventory of these types. Any tips in this regard would be appreciated.

Thanks for your time.

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  • I just realized that negation should be a type of <t,t>.
    – Yili Xia
    Dec 10, 2022 at 11:08
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    I’m afraid I’m not clear what you mean by “What is the type of a purely functional preposition such as of in the picture of John?” Dec 10, 2022 at 21:53

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There are infinite such types because they can be composed with themselves, and you can figure out the type of any word just by observing it.

When you say “As I understand formal semantics so far, every word category should indeed be assigned a type in order for the system to be truly compositional and to hence match the compositional nature of syntactic structures.”, to me it sounds like you are conflating semantics with syntax - you want them to correspond so one-to-one that it almost sounds like you would be trying to make them almost the same thing. I would encourage such an exploration, but what you may not know is that word categories like “verb” and “noun” are often taken as part of the system of syntax, not to be handled a second time within semantics, to an extent. So I don’t think giving syntactic types a corresponding semantic type would be conventional, at least. In semantics there’s actually a famous idea that the “meaning” of words is not related to their syntactic category. I think Aristotle said nouns are things, but it’s more widely held now that words can be changed into different categories - let’s drive - we had a nice drive - the driver - and if you wanted, you could make up the word “drivy”.

It’s a great question but I think you might be overestimating the expressive power of those “types”. The notation is <input, output>, where e is an entity, and t is a proposition. You can figure out the type of any word on your own by considering what complements it takes and what “type” the result is. For example, “is” as a copular verb takes two entities, whereupon it would form a proposition with them: “Jeffrey is cold”. The type is <e, <e, t>>. “To be” can be intransitive, as in the sentence “I think, therefore I am.” In that case, “is” is <e, t>, because it takes an entity and produces a proposition.

This would be my analysis of modal auxiliaries: they usually take a verb as their complement, like “should eat”. The result they return has the same type as the verb it takes.

“I eat bread” -> “I should eat bread”.

“I send it to her” -> “I should send it to her”.

I am pretty sure what this tells us is that technically a modal auxiliary already needs a complete proposition - t - in place, in order to act on it - so my analysis would be that it’s type <t,t>.

The system is described here.

To be honest, there is a great extent to which I have already come to doubt some of the things I said above. One problem is that it is hard to search for this specific type theory convention, it doesn’t appear to have a specific name and I don’t know who invented it. Nevertheless, here’s one reference article that could help one locate the origin of this type theory.

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    Thank you for the answer. As I understand what you write for the sentence Jeffrey is cold, you are suggesting that the copula is is of type <e,<e,t>>. Really? That would mean that the adjective cold is of type e. I think most grammarians view adjectives as predicates, and predicates are generally not interpreted to be individual constants of type e. Perhaps the copula is better analyzed as of type <<e,t>,<e,t>>. This would mean adjectives like cold are of type <e,t>.
    – Yili Xia
    Dec 11, 2022 at 11:21
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    Concerning the sentence I eat bread that you give as an example, the system can, I believe, easily accommodate such sentences. The finite transitive verb eat is of type <e,<e,t>>. No problem. But switching to the sentence I should eat bread, what is the type of the infinitive eat? Is it of the same type as that of finite eat? If yes, then the type of the modal should cannot be of type <t,t> contrary to what you suggest, but rather it would have to be like the copula, that is, it would have to be of type <<e,t>,<e,t>>.
    – Yili Xia
    Dec 11, 2022 at 11:23
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    But I think these matters must have been worked out long ago if the system of types is to be taken seriously. There must be an inventory of these types that is available and accessible. If there is no such inventory, how can one take the discussions of type theory that one encounters in formal semantics seriously?
    – Yili Xia
    Dec 11, 2022 at 11:24

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