13

Nice question, I think this is good to ask for linguistic theory in general, because people who are not so familiar with linguistic research often find this hard to imagine. First of all, logic in general is essential in formal semantics. Using propositional logic, predicate logic, set theory and tools like lambda calculus, functions and type theory, formal ...


8

I can't see the sense in reducing everything in a language to either an "e" or a "t". Maybe this is a good place to start. Type theory (more accurately: so-called simple type theory with e and t as atomic types, which is what most linguists mean when they say "type theory") doesn't reduce everything to either e or t: It reduces ...


7

I really depends on what you are after. Here is a list of my favorite text books, together with some short annotations. Heim & Kratzer 1998: one of the best intro to semantics if you are interested in the interface between syntax and semantics and working a generative grammar background for syntax. Intentionally a bit light on the logical background, ...


6

Upward entailment means that if a relation holds for some set X, then the relation will hold for a superset of X. Downward entailment means that if a relation holds for some X, then the relation will hold for a subset of X. X is a subset of Y and conversely Y is a superset of X iff all elements (members) of X are also elements of Y, i.e. Y "includes" X. ...


6

As the name already suggests, truth conditional semantics is only interested in the truth of a statement, not so much in whether or not that statement makes sense pragmatically. Form a truth-conditional perspective, the sentence "Colorless green ideas sleep furiously" is a syntactically well-formed sentence (not only a formula, but a sentence), and well-...


6

There are fewer academic jobs now compared to the 70's and 80's (in general), and the wave of "baby boom" retirements is just starting. Formal semantics is a "later" field and scholars in that field are comparatively young (in relationship to e.g. syntax). In one sense, there is little demand because it has already been filled. However, formal semantics is ...


5

Here's a short and perhaps inadequate answer: the correspondence is briefly but clearly sketched in the wikipedia article "pregroup grammar". The simplest pop-sci reference I know of is an article from Bob Coecke, in New Scientist. If you stare at the diagram that appears in that article, you will notice it has a very striking resemblance to the link-...


5

(Why is there still no MathJax support for this SE?! Googling and copy-pasting unicode symbols every time you want to talk about semantics is really annoying.) What you want to say is basically ¬φ(a) ∧ φ(b) ("The satement φ involving a is not true, but it is true involving b" = "Unlike a, statement φ applies to b). Note that my a corresponds to your set B ...


5

It's really nice to know that there are some other guys who are interested in type-theoretical semantics. I will give you some advice and references that might be interesting. First, there is a crucial difference between simple type theory that is used in the traditional Montagovian formal semantics (that is, something you would normally find in an ...


4

The expression of family relations does not occur in a social vacuum: it is tied up with the cultural norms surrounding those relations. If you live in a society where the dominant norm is the nuclear family, then the distinction between (1) and (2) is socially fine print: as an adult, you interact with a sister and her husband as a nuclear unit, and with ...


4

I warmly recommend Coppock & Champollion (2020). It's free, very accessibly written and essentially a formally precise version of Heim & Kratzer's style. It also comes with a computer program students can use to get feedback on their solutions to exercises. If you want a rather concise introduction, Winter (2016) might be worth taking a look at. Heim ...


4

I'm surprised you don't mention DRT (Discourse Representation Theory). DRT is a semantic framework that developed, amongst others, from the need to account for anaphoric references (pronouns) across sentence boundaries. As you already suspect, a discourse (roughly what you mean by "paragraph") is seen as a conjunction of individual propositions. DRT is not ...


4

Coppock & Champollion (2021) adopts a formally more rigid version of the Heim & Kratzer style and covers a number of advanced topics. Intermediate-level formal semantics with less syntax and more logic can be found in Gamut (1991) (covering intensional logic and Montague Grammar) and Gregory (2015) (dealing some more with proof systems). A different ...


3

Theorems are wffs, but not all wffs are theorems. Theorems are demonstrable, but other wffs are not. The central notion in logic is implication, not truth. It is possible to answer questions about implication without making any appeal to truth -- a logic constructed that way is a "logical syntax". If there is appeal to truth, in a "logical semantics", it ...


3

While the definite article is assumed to exhibit both an existence and a uniqueness presupposition, combining to an "exactly one" presupposition: I saw the bear yesterday → There is a bear → There is not more than one bear → = There is exactly one bear (in this situation, e.g. in a town) this is generally not assumed for the indefinite article: ...


3

When we implemented our software for 'paragraph analysis' it ended up looking like a 'union of propositions' with two minor changes. 1. paragraphs were often preceded by headings which we used to set the context of the paragraph (like a topic model/domain) 2. We used a modification of this simple framework: http://depts.washington.edu/pswrite/Handouts/...


3

Type theory avoids the paradoxes in set theory that were discovered early in the last century, e.g. Russell's paradox. It is not the only way to avoid the paradoxes. It is often used in the formal development of higher order logics. I am not aware of any reason to use it for linguistics, other than the popularity of Montague grammar, which does use higher ...


3

To express "some students" in the sense of "more than one", you could say that there exist at least two distinct individuals both of which are students and met by John (and yes, the duplication is necessary): ∃x ∃y (x ≠ y ⋀ student(x) ⋀ student(y) ⋀ met(j,x) ⋀ met(j,y)) "at least one but not all" would be "There is at ...


2

I'm not sure I entirely got the point (I find it weird to say that the variable can be eliminated in natural language, because 1. you don't explicitly use variables in natural language anyway and 2. "there is some fox..." is not a variable, so I find that wording misleading), but I think it's important for you to have the the basic definitions in mind (sorry ...


2

Judging from your comments above, I'm guessing that your task is to show how the principle of compositionality applies to the interpretation of your example "a former president with red hair". If so, it was hard to figure that out, because you're using the terms "proof" and "explain" in an odd way. I don't see how "proof" comes into it -- rather, you seem ...


2

I think perhaps with an qualifying adjective. "Are you a good mother?" Presupposes that the referent of "you" is, in fact, a mother.


2

a) Correct. Some fine-tuning to your answer: The donkey that Jake owns in own(j,x), is not the same donkey that was beaten in beat(j,x). It is not necessarily the same. Depending on which assignment function the formula is evaluated under, the assignment for the second occurence of the variable x can co-incide with the one for the first x. But what is ...


2

The reason is as follows: For ALL we need "for all objects, if they are P, then they are also Q". If we would use logical AND, it would mean "for all objects, they are P and they are Q" which is obviously not what we want. For SOME we need "for some object, it is P and it is Q". If we would use implication, it would mean "for some object, if it is P, then ...


2

The wording of your question seems to imply an equivalence between first order predicate logic and higher order logics. They are not equivalent. First order predicate logic was shown to be consistent and complete by Kurt Gödel, but "Stronger logics, such as second-order logic, are not complete." (Consistency.) Whether first order predicate logic is ...


2

I don't see the problem. If you know how to handle restrictive relative clauses, then you must know how to handle adverbial clauses. The example "you can sit where you like" means "you can sit in any place in which you like to sit". Except for some superficial syntax, the "adverbial clause" is a restrictive relative already. (Jonnie and Mike Geis both ...


2

It is a relative clause. A plot of land is acting as the lexical head of the RC, and in relation to was 'moved' to before the relative pronoun (not conjunction) which through the process of pied piping; you can see more examples on the Wikipedia page linked.


2

Yes. See Arc pair grammar....


2

I think the best way to understand in detail how the formalism (Minimal Recursion Semantics) works is simply to have a look at the original paper by Copestake et.al. You may want to skip the motivation and formal definitions and focus on the representation and the implementation into feature structures/HPSG. Maybe this summary helps for an overview, although ...


2

This refers to the existential quantifier, "exists X", and universal quantifier, "for all x", respectively in first order logic, which quantifies formulas over the elements of a set, not to be confused with higher order logic, that quantifies formulas over classes of formulas. The donkey-beating sentences are the canonical example in this context in ...


2

Yes, it’s useful. Formal semantics can serve as a basis for the stochastic methods. There are many approaches, let me just mention one — abductive parsing and interpretation. It’s based on formal semantics but the algorithms for analysing text are stochastic. IBM Watson is an example of a system based on formal semantics.


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