Formal semantics (natural language)

Formal semantics is the scientific study of grammatical meaning in natural languages using formal concepts from logic, mathematics and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. It provides accounts of what linguistic expressions mean and how their meanings are composed from the meanings of their parts. The enterprise of formal semantics can be thought of as that of reverse-engineering the semantic components of natural languages' grammars.

Definition

Formal semantics is an approach to the study of linguistic meaning that uses ideas from logic and philosophy of language to characterize the relationships between expressions and their denotations. These tools include the concepts of truth conditions, model theory, and compositionality.

Formal semantics is related to formal pragmatics since both are subfields of formal linguistics. One key difference is that formal pragmatics centers on how language is used in communication rather than the problem of meaning in general. Formal semanticists examine a wide range of linguistic phenomena, including reference, quantifiers, plurality, tense, aspect, vagueness, modality, scope, binding, conditionals, questions, and imperatives.

Formal semantics is an interdisciplinary field, often viewed as a subfield of both linguistics and philosophy, while also incorporating work from computer science, mathematical logic, and cognitive psychology. Formal semanticists typically adopt an externalist view of meaning that interprets meaning as the entities to which expressions refer. This focus on the connection between language and the external world sets formal semantics apart from semantic theories that concentrate on the cognitive processes and mental representations involved in understanding language.

The primary focus of formal semantics is the analysis of natural language such as English, Spanish, and Japanese. This enterprise faces challenges due to the complexity and context-dependence of natural language. As a result, theorists sometimes limit their studies to specific fragments or subsets of these languages to avoid these complexities. Understood in a wide sense, formal semantics also includes the study of artificial or constructed languages. This covers the formal languages used in the logical analysis of arguments, such as the language of first-order logic, and programming languages in computer science, such as C++, JavaScript, and Python.

Methodology

Formal semanticists rely on diverse methods, conceptual tools, and background assumptions, which distinguish the field from other branches of semantics. Most of these principles originate in logic, mathematics, and the philosophy of language. One key principle is that an adequate theory of meaning needs to accurately predict sentences' truth conditions. A sentence's truth conditions are the circumstances under which it would be true. For example, the sentence "Tina is tall and happy" is true if Tina has the property of being tall and also the property of being happy; these are its truth conditions. This principle reflects the idea that understanding a sentence requires knowing how it relates to reality and under which circumstances it would be appropriate to use it.

In formal semantics, a key source of data comes from competent speakers' judgments concerning entailment. Entailment is a relation between sentences—called premises and conclusions—in which truth is preserved. For instance, the sentence "Tina is tall and happy" entails the sentence "Tina is tall" because the truth of the first sentence guarantees the truth of the second. One aspect of understanding the meaning of a sentence is comprehending what it does and does not entail.

Formal semanticists typically analyze languages by proposing a system of model theoretic interpretation. In such a system, a sentence is evaluated relative to a mathematical structure called a model that maps each expression to an abstract object. For instance, one model might map the expression "tall" to the set consisting of Tina, Bob, and Jeroen and the predicate "happy" to the set consisting of Tina, Grover, and Natasha. A different model might map those expressions to the opposite sets. In this way, different models can be used to represent different situations. Whichever model is chosen, "Tina is tall" will be true if and only if she is in the set to which that model assigns the predicate "tall". In this way, model theoretic interpretation provides a way of formally capturing sentences' truth conditions and thus their entailments as well.

The principle of compositionality is another key methodological assumption for analyzing the meaning of natural language sentences and linking them to abstract models. It states that the meaning of a compound expression is determined by the meanings of its parts and the way they are combined. According to this principle, if a person knows the meanings of the name Tina, the verb is, and the adjective happy, they can understand the sentence "Tina is happy" even if they have never heard this specific combination of words before. The principle of compositionality explains how language users can comprehend an infinite number of sentences based on their knowledge of a finite number of words and rules. Following this principle, formal semanticists connect natural language sentences to abstract models through a form of translation, for instance, by defining an interpretation function that maps the name "Tina" to an abstract object and the adjective "happy" to a set of objects. This makes it possible to precisely calculate the truth values of sentences relative to abstract models.

Formal systems and theories

Propositional and predicate logic

Main articles: Propositional logic and Predicate logic

Formal semanticists often rely on propositional and predicate logic to analyze the semantic structure of sentences. Propositional logic can be used to examine compound sentences made up of several independent clauses. It employs letters like $A$ and $B$ to represent simple statements. Compound statements are created by combining simple statements with logical connectives, such as $\land$ (and), $\lor$ (or), and $\to$ (if...then), which express the relationships between the statements. For example, the sentence "Alice is happy and Bob is rich" can be translated into propositional logic with the formula $A\land B$, where $A$ stands for "Alice is happy" and $B$ stands for "Bob is rich". Of key interest to semantic analysis is that the truth value of these compound statements is directly determined by the truth values of the simple statements. For instance, the formula $A\land B$ is only true if both $A$ and $B$ are true; otherwise, it is false.

Predicate logic extends propositional logic by articulating the internal structure of non-compound sentences through concepts like singular term, predicate, and quantifier. Singular terms refer to specific entities, whereas predicates describe characteristics of and relations between entities. For instance, the sentence "Alice is happy" can be represented with the formula $Happy(alice)$, where $alice$ is a singular term and $Happy$ is a predicate. Quantifiers express that a certain condition applies to some or all entities. For example, the sentence "Someone is happy" can be represented with the formula $\exists xHappy(x)$, where the existential quantifier $\exists$ indicates that happiness applies at least to one person. Similarly, the idea that everyone is happy can be expressed through the formula $\forall xHappy(x)$, where $\forall$ is the universal quantifier.

There are different ways how natural language sentences can be translated into predicate logic. A common approach interprets verbs as predicates. Intransitive verbs, like sleeps and dances, have a subject but no objects and are interpreted as one-place predicates. Transitive verbs, like loves and gives, have one or more objects and are interpreted as predicates with two or more places. For example, the sentence "Bob loves Alice" can be formalized as $Loves(bob,alice)$, using the two-place predicate $Loves$. Typically, not every word in natural language sentences has a direct counterpart symbol in the logic translation, and in some cases, the pattern of the logical formula differs significantly from the surface structure of the natural language sentence. For example, sentences like "all cats are animals" are usually translated as $\forall x(Cat(x)\to Animal(x))$ (for all entities, if the entity is a cat then the entity is an animal) even though the expression "if...then" ($\to$) is not present in the original sentence.

Logic translations face challenges as a result of attempting to associate vague and ambiguous ordinary language expressions with precise logical formulas. This process frequently requires case-by-case interpretation without a generally accepted algorithm to cover all cases. Many early approaches to formal semantics, such as the works of Gottlob Frege, Rudolf Carnap, and Donald Davidson, relied primarily on predicate logic.

Type theory

Main article: Type theory

Type theory is another approach to formal semantics that was popularized by Montague. Its core idea is that expressions belong to different types, which describe how the expressions can be used and combined with other expressions. Type theory, typically in the form of typed lambda calculus, provides a formalism for this endeavor. It begins by defining a small number of basic types, which can be fused to create new types.

According to a common approach, there are only two basic types: entities ($e$) and truth values ($t$). Entities are the denotations of names and similar noun phrases, while truth values are the denotations of declarative sentences. All other types are constructed from these two types as functions that have entities, truth values, or other functions as inputs and outputs. This way, a sentence is analyzed as a complex function made up of several internal functions. When all functions are evaluated, the output is a truth value. Simple intransitive verbs without objects are functions that take an entity as input and produce a truth value as output. The type of this function is written as $\langle e,t\rangle$, where the first letter indicates the input type and the second letter the output type. According to this approach, the sentence "Alice sleeps" is analyzed as a function that takes the entity Alice as input to produce a truth value. Transitive verbs with one object, such as the verb likes, are complex or nested functions. They take an entity as input and output a second function, which itself requires an entity as input to produce a truth value, formalized as $\langle e,\langle e,t\rangle \rangle$. This way, the sentence "Alice likes Bob" corresponds to a nested function to which two entities are applied. Similar types of analyses are provided for all relevant expressions, including logical connectives and quantifiers.

Others

Possible worlds are another central concept used in the analysis of linguistic meaning. A possible world is a complete and consistent version of how everything could have been, similar to a hypothetical alternative universe. For instance, the dinosaurs were wiped out in the actual world but there are possible worlds where they survived. Possible worlds have various applications in formal semantics, usually to study expressions or aspects of meaning that are difficult to explain when referring only to entities of the actual world. They include modal statements about what is possible or necessary and descriptions of the contents of mental states, such as what people believe and desire. Possible worlds are also used to explain how two expressions can have different meanings even though they refer to the same entity, such as the expressions "the morning star" and "the evening star", which both refer to the planet Venus. One way to include possible worlds in the model-theoretic formalism is to define a set of all possible worlds as one additional component of a model. The interpretation of the meanings of different expressions is then modified to account for this change. For example, to explain that a sentence may be true in one possible world and false in another, one can interpret its meaning not directly as a truth value but as a function from a possible world to a truth value.

Situation semantics is a theory closely related to possible world semantics. Situations, like possible worlds, present possible circumstances. However, unlike possible worlds, they do not encompass a whole universe but only capture specific parts or fragments of possible worlds. This modification reflects the observation that many statements are context-dependent and aim to describe the speaker's specific circumstances rather than the world at large. For example, the sentence "every student sings" is false when interpreted as an assertion about the universe as a whole. However, speakers may use this sentence in the context of a limited situation, such as a specific high school musical, in which it can be true.

Dynamic semantics interprets language usage as a dynamic process in which information is continually updated against the background of an existing context. It rejects static approaches that associate a given expression with a fixed meaning. Instead, this theory argues that meaning depends on the information that is already present in the context, understanding the meaning of a sentence as the change in information it produces. This view reflects the idea that sentences are usually not interpreted in isolation but form part of a larger discourse, to which they contribute in some way. For example, update semantics—one form of dynamic semantics—defines an information state as the set of all possible worlds compatible with the current information, reflecting the idea that the information is incomplete and cannot determine which of these worlds is the right one. Sentences introducing new information update the information state by excluding some possible worlds, thereby decreasing uncertainty.

Studied linguistic phenomena

Quantifiers

Main article: Generalized quantifier

Quantifiers are expressions that indicate the quantity of something. In predicate logic, the most basic quantifiers only provide information about whether a condition applies to all or some entities, as seen in sentences like "all ravens are black" and "some students smoke". Formal semanticists use the concept of generalized quantifiers to extend this basic framework to a broad range of quantificational expressions in natural language that usually provide more detailed information. They include diverse expressions such as most, few, twelve, and fewer than ten.

Most quantificational expressions can be interpreted as relations between two sets. For instance, the sentence "all ravens are black" conveys the idea that the set of ravens is a subset of the set of black entities. Similarly, the sentence "fewer than ten books were sold" asserts that the set of books and the set of sold items have fewer than ten elements in common. In English, quantifiers are often expressed with a determiner, such as all and few, indicating the relation between the sets, followed by a noun phrase and a predicate to describe the involved sets.

Quantifiers can be divided into proportional and cardinal quantifiers based on the relation between the sets. Proportional quantifiers, such as all and most, indicate the relative overlap of the first set with the second set. For them, the order of the sets matters. For instance, the sentences "all ravens are black" and "all black things are ravens" have different meanings even though they refer to the same sets. Cardinal quantifiers, such as four and no, provide information about the absolute number of overlapping entities, independent of relative proportion. For them, the order of the sets does not matter, as exemplified by the sentences "no rose is black" and "no black thing is a rose".

Typically, the domain of natural language quantifiers is implicitly limited to a certain range of entities relevant to the discussed issue. For example, in the context of a specific kindergarten, the domain of the sentence "all children are sleeping" is limited to the children attending this kindergarten.

The scope of a quantifier is the part of the sentence to which it applies. Some natural language sentences have scope ambiguity, resulting in competing interpretations of the scope of quantifiers. Depending on how the scope is interpreted, the sentence "Some man loves every woman" can mean either "there is a man such that he loves all women" or "for every woman there is at least one man who loves her".

Descriptions and names

Main article: Theory of descriptions

Definite and indefinite descriptions are phrases that denote a specific entity or group of entities within a given context. Definite descriptions in English typically use the definite article the, followed by a noun phrase, such as "the president of Kenya". However, they can also take other forms, such as "her husband" or "John's bicycle". Indefinite descriptions are usually expressed with the indefinite articles a and an, as in a lazy coworker and an old friend. Definite descriptions typically point to a unique entity and assume that the listener is familiar with the referent. Indefinite descriptions usually allow the description to apply to more than one entity and introduce the entity without presupposing prior knowledge.

Diverse theories about the correct analysis of definite and indefinite descriptions have been proposed. An influential early view, suggested by Bertrand Russell, interprets them using existential quantifiers. It proposes that indefinite descriptions like "a man ran" have the logical form $\exists x(Man(x)\land Ran(x))$. Definite descriptions have a similar form, with the difference that the description is unique, meaning that the first predicate only applies to a single entity. A central motivation for Russell's approach was to solve semantic puzzles that arise from definite descriptions that do not refer to any particular entity. For example, the sentence "the present king of France is bald" refers to no existing entity, posing challenges for determining its truth value. According to Russell's analysis, the sentence is false since no unique entity exists to which the predicates "present king of France" and "bald" apply.

The problem of names is closely related to that of definite descriptions because both expressions aim to refer to a particular entity. According to Millian theories, names refer directly without any descriptive information of the denoted entity. This view is opposed by description theories, which argue that names carry implicit descriptive contents that help interpreters identify their referents. One view understands names as implicit definite descriptions, proposing that the descriptive content of the name Socrates may include information like "the teacher of Plato".

Tense, aspect, and events

Tense and aspect provide temporal information about events and circumstances. Tense indicates whether something happened in the past, present, or future, offering a reference point to place events within a timeline relative to the time of the utterance. Aspect conveys additional information about how events unfold in time, like the distinction between completed, ongoing, and repetitive events. In English, both tense and aspect can be expressed through verb forms. On the level of tense the sentence "I ate" indicates the past, whereas the sentence "I will eat" indicates the future. On the level of aspect, the sentence "I ate" indicates a completed action, whereas the sentence "I was eating" indicates an ongoing action.

Formal semanticists employ diverse conceptual tools to describe tense, such as different types of temporal logic as extensions of predicate logic. One approach includes a set of times in the mathematical model to interpret temporal statements. Some models conceptualize time as a series of instances, while others introduce intervals as the basic units of time. The difference is that intervals have a duration and can overlap, whereas instances are discrete time points that do not intersect. One form of temporal logic introduces tense operators to indicate the time a sentence describes, like the operator $P$ for past events and the operator $F$ for future events. This way, the formula $P\ Dance(naomi)$ expresses that Naomi danced in the past, while $F\ Dance(naomi)$ asserts that she will dance in the future. The semantic analysis of aspect is divided into grammatical aspect, expressed through verb forms, and lexical aspect, which covers the inherent temporal characteristics of different verbs.

An influential approach to the semantic role of events was proposed by Donald Davidson. Using predicate logic, it represents events as singular terms and translates action sentences into logical formulas about events, even if the original sentences contain no explicit reference to events. For example, it translates the sentence "Jones buttered the toast slowly with a knife" as $\exists eButter(Jones,the\ toast,e)\land Slowly(e)\land With(e,a\ knife)$ (literally: there was an event, which was a buttering of the toast by Jones, was slow, and involved a knife). One motivation for this approach is to provide a systematic method for translating adverbs, like slowly, and other adjuncts into logical formulas.

Intensionality, modality, and propositional attitudes

Main articles: Intensionality, Linguistic modality, and Propositional attitude

Semanticists often distinguish two aspects of meaning: extension and intension. Extension is the entity or group of entities to which an expression refers, while intension is the inherent concept or underlying idea it conveys. For example, the expressions "the morning star" and "the evening star" have the same extension, as both refer to the planet Venus. However, their meanings differ on the level of intension since they present the planet in different ways by evoking distinct concepts.

Extensionality and intensionality are characteristics of sentences. A sentence is extensional if expressions with the same extension can be substituted without changing the sentence's truth value. For example, the sentence "the morning star is a planet" remains true if the expression "the morning star" is replaced with the expression "the evening star". Intensional sentences, by contrast, are not only sensitive to extensions but also to intensions, meaning that extensionally equivalent expressions cannot be freely replaced. For instance, the sentence "Ann knows that the morning star is the morning star" is intensional since it can be true while the extensionally equivalent sentence "Ann knows that the evening star is the morning star" is false.

Intensionality is present in various linguistic expressions. For instance, modal expressions, such as may, can, and must, usually introduce intensional contexts. They express what is possible or necessary, describing how the world could or could not have been rather than how it actually is. A common approach to the analysis of modal expressions is the use of the modal operators $\Diamond$ and $\Box$ to modify the meaning of sentences and represent what is possible and necessary. For example, if the formula $P$ stands for the statement "it is raining", then the formula $\Diamond P$ stands for the statement "it is possible that it is raining". To interpret the meaning of modal statements, formal semanticists often rely on the concept of possible worlds. According to this approach, a sentence is possibly true if it is true in at least one possible world, whereas it is necessarily true if it is true in all possible worlds.

Propositional attitude reports—another example of intensionality—discuss mental states of individuals. They often use verbs like believes, doubts, and wants, followed by a that-clause describing the content of the attitude, like the sentence "Kyrie believes that the earth is flat". The use of possible worlds is also common for the analysis of propositional attitudes. For example, the content of a propositional attitude can be understood as the set of all possible worlds in which it is true, such as all possible worlds with a flat earth in the mentioned example. The meaning of propositional attitude reports containing definite or indefinite descriptions is often ambiguous. This ambiguity arises from the interpretation of the description, which can be subjective or objective. For example, if Jasper wants a drink from his butler but is unaware that his butler poisoned his wife, then the sentence "Jasper wants a drink from the poisoner of his wife" is ambiguous. According to the objective interpretation—called de re interpretation—the sentence is true since the butler is in fact the poisoner. Conversely, the subjective interpretation—called de dicto interpretation—renders it false since Jasper does not want drinks from poisoners.

Questions and imperatives

Main articles: Question and Imperative mood

The main focus of formal semantics is on statements, which aim to describe reality and are either true or false depending on whether they succeed. However, this analysis does not cover all types of sentences. Specific frameworks have been proposed for the analysis of other sentence types, such as questions and imperatives.

Various theories analyze the meaning of questions in terms of possible answers, replacing the concept of truth conditions, common in the analysis of statements, with the related notion of answerhood conditions. One approach, initially formulated by Charles Leonard Hamblin, interprets answerhood conditions as the set of statements that qualify as answers to a question. For instance, the sentences "Marco called" and "Don called" qualify as answers to the question "Who called?", but the sentence "I like ice cream" does not. A common distinction is between yes-no questions, which only ask for confirmation, and open-ended questions, which seek more detailed information. Additional considerations include the distinctions between true and false answers, and between complete and partial answers, depending on whether the response contains all the requested information. On the symbolic level, questions can be expressed using $?$ as an operator to indicate the subject of the question. For example, the question "Who called?" can be formalized as $?xCalled(x)$, whereas the question "Did anyone call?" takes the form $?\exists xCalled(x)$.

Imperative sentences usually express commands or instructions, like the sentence "Close the door!". Unlike declarative and interrogative sentences, which generally convey or request information, the primary goal of imperatives is to influence the behavior of the listener. As a result, imperatives have no or at least no obvious truth conditions. Other difficulties in the analysis of imperative sentences are that they usually lack an explicit subject and that they can express various other meanings besides commands, such as advice, invitations, or permissions. Formal semanticists study the meaning of imperatives by examining how they interact with other linguistic phenomena. These include cases in which one imperative entails another imperative, the negation of an imperative, and conditional imperatives as well as conjunctions and disjunctions of several imperatives.

Others

Diverse other linguistic phenomena are studied in formal semantics. Negation is typically understood as an operation that inverts the meaning of an expression. In classical logic, it is expressed through the operator $\lnot$ as in $\lnot Sleeps(mia)$, indicating that Mia is not sleeping. This operator inverts the truth value of a statement: if $Sleeps(mia)$ is false then $\lnot Sleeps(mia)$ is true. In natural language, negative particles and quantifiers, such as not and no, are often used to indicate negation. These expressions can occur in different positions within sentences to negate either the full sentence or specific parts of it. The scope of a negation operator is the part of the sentence that it affects, which can sometimes be ambiguous. For example, the sentence "all doctors have no car" can mean that not every doctor has a car, that not a single doctor has a car, or that no individual car is collectively owned by all doctors.

Plural expressions refer to multiple objects, such as the terms children and apples. Formal semanticists typically interpret them as denoting some kind of plural object, such as the set of individuals belonging to the group in question. They distinguish between distributive and collective uses depending on whether the predicate applies to each individual separately or to the group as a whole. Some sentences are ambiguous and allow for both interpretations. For example, the sentence "two boys pushed a car" can mean that there were two cars and each boy pushed one (distributive) or that there was one car that both boys pushed together (collective).

Formal semanticists also examine expressions whose meaning depends on contextual factors. They include indexical or deictic expressions, which refer to some aspect of the situation of the text. Examples are the pronouns I and you, which refer to the speaker and the addressee, as well as the adverbs today and over there, which refer to temporal and spatial aspects of the situation. Anaphoric expressions are another type of context-dependent expression. They refer to terms or phrases used earlier in the text, called antecedents. In the passage "Peter woke up. He switched on the light." the word he is an anaphoric expression with the word Peter as its antecedent. This grammatical association is known as binding and depends on the context since the word he would refer to someone else if the preceding sentence had a different antecedent. Other linguistic phenomena studied by formal semanticists include presupposition, conditionals, thematic roles, spatial expressions, adjectives, and adverbs.

History

Formal semantics emerged as a major area of research in the early 1970s, with the pioneering work of the philosopher and logician Richard Montague. Montague proposed a formal system now known as Montague grammar which consisted of a novel syntactic formalism for English, a logical system called Intensional Logic, and a set of homomorphic translation rules linking the two. In retrospect, Montague Grammar has been compared to a Rube Goldberg machine, but it was regarded as earth-shattering when first proposed, and many of its fundamental insights survive in the various semantic models which have superseded it.

Barbara Partee is one of the founders and major contributors to the field.

Montague Grammar was a major advance because it showed that natural languages could be treated as interpreted formal languages. Before Montague, many linguists had doubted that this was possible, and logicians of that era tended to view logic as a replacement for natural language rather than a tool for analyzing it. Montague's work was published during the Linguistics Wars, and many linguists were initially puzzled by it. While linguists wanted a restrictive theory that could only model phenomena that occur in human languages, Montague sought a flexible framework that characterized the concept of meaning at its most general. At one conference, Montague told Barbara Partee that she was "the only linguist who it is not the case that I can't talk to".

Formal semantics grew into a major subfield of linguistics in the late 1970s and early 1980s, due to the seminal work of Barbara Partee. Partee developed a linguistically plausible system which incorporated the key insights of both Montague Grammar and Transformational grammar. Early research in linguistic formal semantics used Partee's system to achieve a wealth of empirical and conceptual results. Later work by Irene Heim, Angelika Kratzer, Tanya Reinhart, Robert May and others built on Partee's work to further reconcile it with the generative approach to syntax. The resulting framework is known as the Heim and Kratzer system, after the authors of the textbook Semantics in Generative Grammar which first codified and popularized it. The Heim and Kratzer system differs from earlier approaches in that it incorporates a level of syntactic representation called logical form which undergoes semantic interpretation. Thus, this system often includes syntactic representations and operations which were introduced by translation rules in Montague's system. However, work by others such as Gerald Gazdar proposed models of the syntax-semantics interface which stayed closer to Montague's, providing a system of interpretation in which denotations could be computed on the basis of surface structures. These approaches live on in frameworks such as categorial grammar and combinatory categorial grammar.

Cognitive semantics emerged as a reaction against formal semantics, but there have been recently several attempts at reconciling both positions.

See also

• Alternative semantics
• Compositionality
• Computational semantics
• Discourse representation theory
• Dynamic semantics
• Frame semantics (linguistics)
• Inquisitive semantics
• Philosophy of language
• Pragmatics
• Semantics of logic
• Syntax–semantics interface
• Traditional grammar

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Further reading

• Max Cresswell (2006). "Formal semantics". In Michael Devitt, Richard Hanley (ed.). The Blackwell guide to the philosophy of language. Wiley-Blackwell. ISBN 978-0-631-23142-4. A very accessible overview of the main ideas in the field.
• John I. Saeed (2008). Semantics. Introducing linguistics (3rd ed.). Wiley-Blackwell. ISBN 978-1-4051-5639-4. Chapter 10, Formal semantics, contains the best chapter-level coverage of the main technical directions
• Johan van Benthem; Alice Ter Meulen (2010). Handbook of Logic and Language (2nd ed.). Elsevier. ISBN 978-0-444-53726-3. The most comprehensive reference in the area.
• Emmon W. Bach (1989). Informal lectures on formal semantics. SUNY Press. ISBN 978-0-88706-772-3. One of the first textbooks. Accessible to undergraduates.
• Ronnie Cann (1993). Formal semantics: an introduction. Cambridge University Press. ISBN 978-0-521-37610-5.
• Irene Heim; Angelika Kratzer (1998). Semantics in generative grammar. Wiley-Blackwell. ISBN 978-0-631-19713-3.
• Gennaro Chierchia; Sally McConnell-Ginet (2000). Meaning and grammar: an introduction to semantics (2nd ed.). MIT Press. ISBN 978-0-262-53164-1.
• Sean A. Fulop (2004). On the Logic and Learning of Language. Trafford Publishing. ISBN 978-1-4120-2381-8.
• Glyn V. Morrill (1994). Type logical grammar: categorial logic of signs. Springer. ISBN 978-0-7923-3095-0.
• Reinhard Muskens. Type-logical Semantics. Routledge Encyclopedia of Philosophy Online.
• Bob Carpenter (1998). Type-logical semantics. MIT Press. ISBN 978-0-262-53149-8.
• Johan van Benthem (1995). Language in action: categories, lambdas, and dynamic logic. MIT Press. ISBN 978-0-262-72024-3.
• Barbara H. Partee. Reflections of a formal semanticist as of Feb 2005. Ample historical information. (An extended version of the introductory essay in Barbara H. Partee: Compositionality in Formal Semantics: Selected Papers of Barbara Partee. Blackwell Publishers, Oxford, 2004.)