Formal models in linguistics: semantics

Author: David Adger


Teaching formal semantics: an outline of the core issues and some possible approaches

Table of contents


Semantics is the study of meaning in natural languages. Formal Semantics is a discipline that has its roots in logic, philosophy and linguistics. It approaches questions of meaning from a particular perspective: rather than taking the relationship of language to thought, or language to communication as crucial for understanding meaning, it takes the relationship between language and truth as primary. This is one particular way of grounding the notion of meaning: if we take two sentences, and can imagine a situation where one is true and one is false, then we can be certain that they have different meanings. If we rely, instead, on intuitions about them expressing the same thought, or bearing the same message, then we are on much shakier ground. Formal semantics proceeds on the assumption that taking truth as primary gives us more stable foundation than would otherwise be possible (at least given our current state of knowledge about thought and communication).

Because formal semantics takes truth as the core concept relevant to meaning, the basic data of the discipline are truth conditions and entailments. The truth conditions for a sentence are the conditions that would have to hold in the world for the sentence to be true, and one sentence entails another if there is no way for the first to be true and the second false. Taking these basic facts about human language as the data for investigation leads to a very particular (and extremely successful) way of understanding linguistic meaning.

When teaching formal semantics, it is important to make a clear distinction between three aspects of the field: the way that semantic representations are built up; the mapping of these semantic representations to non-linguistic representations, and the nature of the non-linguistic representations. Most semantics courses focus on the first two tasks, leaving the last one (which is essentially the research agenda outlined in Bach’s (1986) Natural Language Metaphysics) to more advanced levels).

Let us focus on the first task: how semantic representations are built up. It is of course possible to do formal semantics without adopting any particular semantic representation (in fact, this is an approach adopted by Montague in some of his work; see Dowty et al (1981) for discussion); however, a core skill for students learning to do semantics is being able to paraphrase meanings of sentences using an explicit formal language. This is the best way to make truth conditions explicit. For this reason, it is possible to teach a course in formal semantics which essentially gets students to learn some version of predicate logic, and then to represent sentences of natural language using the language of logic (see Guttenplan (1997) for a particularly good text). However, this is to miss out on one of the great successes of formal semantics, which is the compositional construction of the meanings of whole sentences on the basis of the meanings of their parts and the ways in which their parts are combined. To do this properly, students need to be exposed not only to a formal semantic representation language, but also to ways of combining the constituent parts of that language.

There are again a number of ways to do this: one might use predicate logic, augmented with the lambda-operator; or one might adopt a typed version of logic augmented with the lambda-operator. The advantage of the latter approach is that it provides a much more transparent mapping between syntactic structures and the semantic representation, and involves less ‘unlearning’ on the part of the student.

The second task has to do with mapping the compositionally constructed semantic representation to a non-linguistic representation. This is usually done by defining a model which consists of (at least) individuals and truth values, and by defining a mapping between predicates of the language, and individuals, or groupings of individuals, in the model. In addition, a model theory will be assumed, which will provide meanings for truth functional operators such as negation and conjunction. The model and its associated mapping can be constructed in terms of individuals, ordered n-tuples of these, and sets of individuals and n-tuples, or in terms of characteristic functions of sets. The latter approach again makes the mapping between the structure of natural language and that of the model more transparent.

One of the recurrent difficulties in teaching formal semantics to students in linguistics (who often have a languages rather than a maths background) is formalism. Students not only need to get to grips with some fairly abstract conceptual arguments, but they also need to acquire a facility with the appropriate formalism, and also to be able to do some actual linguistics. This is why students find the subject hard, and need a lot of support during the course. However, the results can be extremely rewarding, and a solid background in formal semantics can be a huge help to students of syntax and pragmatics, as well as providing an impressive range of transferable intellectual skills.

There are a number of good texts in formal semantics. One classic is Dowty et al (1981), which leads the reader through an exposition of Montague’s approach. More up to date is the excellent Heim and Kratzer (1998) (marred only by the lack of a proper bibliography), which is especially good if students are being exposed to a transformational approach to syntax in other classes. It is, however, rather brief on the formal foundations needed. Recently, a book of readings has appeared (Portner and Partee 2002) which will be extremely useful for advanced classes.


Allwood, J., L-G Andersson & O. Dahl (1977) Logic in Linguistics, CUP

Bach, E.(1989) Informal Lectures on Formal Semantics. SUNY Press, New York.

Bach, Emmon (1986) Natural Language Metaphysics. In R. Barcan Marcus, G.J.W. Dorn and P. Weingartner (eds) Logic, Methodology and Philosophy of Science VII, 573-595, Amsterdam, North-Holland.

Cann, R. (1993) Formal Semantics: an introduction. CUP; Cambridge.

Chierchia, G. & S. McConnell-Ginet (2000) Meaning and Grammar. MIT Press; Cambridge, MA. (second edition)

Dowty, D., R. Wall and S. Peters (1981) Introduction to Montague Semantics, Reidel, Dordrecht. (the 1985 edition contains some corrections)

Guttenplan, S (1997) The Languages of Logic, Blackwell; Oxford (second edition).

Heim, I & A. Kratzer (1997) Semantics in Generative Grammar, MIT Press, Cambridge, MA.

Larson, R & G. Segal (1995) Knowledge of Meaning, MIT Press, Cambridge, MA.

Partee, B., A. ter Meulen & B. Wall (1990) Mathematical methods in Linguistics, Kluwer.

Portner, P. and B. H. Partee (eds.) (2002) Formal Semantics: The Essential Readings. Blackwell.

De Swart, H. (1998) Introduction to Natural Language Semantics. CSLI: Stanford.

Related links

A good collection of links can be found at the following URL:

Alliance for Lifelong Learning

The SemanticsArchive

Language, Logic, Information

An excellent collection of semantics links:

Semantics Links

Dr Ronnie Cann’s article on Semantics

'Teaching formal semantics'

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