Formal semantics is a broad linguistic theory, intended to study how language functions and its relationship with reality. Linguists carefully define every statement’s truth conditions, which are those situations which must be fact in order for the sentence to be true. Sentences are then transcribed into mathematical equations, primarily using typed lambda calculus.
The theory of formal semantics was first developed by American philosopher and mathematician Richard Montague in the 1960s. His specific version is known as Montague Grammar. Since that time, this theory has developed in many directions, including categorical grammar, as pioneered by Yehoshua Bar-Hillel in the 1970s, and Glue semantics, published by Mary Dalrymple in the 1990s.
Another type, discourse representation theory, was developed in 1981 by Dutch linguist Hans Kamp. This specific linguistic theory is sometimes referred to under the generic term formal semantics. Discourse representation theory attempts to analyze the entire discourse, or conversation, rather than isolated sentences, and present it in mathematical equations.
All branches of formal semantics focus on the relationship between language and reality; in other words, their goal is to find the connection between what is said and what is. Words are referred to as signifiers while the things themselves are called the denotata, or, in discourse representation theory, discourse referents. In the sentence, “The dog barked,” the word “dog” is a signifier, while the actual dog is a discourse referent.
A statement may have more than one truth condition, or requirement in order to be true. Often these truth conditions lead to entailment, which is the term used when one sentence’s truth requires another to be true. In other words, in an entailed situation, if sentence A is true, then sentence B must be true as well.
To determine a statement’s truth conditions, linguists must look for any quantifiers present. A quantifier is a word which indicates how many things are involved, such as each, every, any, and some. These words can change a statement’s meaning substantially.
The type of mathematics most often used to analyze language in formal semantics is typed lambda calculus. Words can be identified as different variables and placed within an equation; in discourse representation theory, these equations are known as discourse representation structures. Although primarily a linguistic theory, formal semantics is a multidisciplinary field. Linguists, philosophers, logicians, mathematicians and programmers are all involved. Analyzing and creating programming languages and even studying artificial intelligence can all involve formal semantics.