User:Phlsph7/Formal semantics - Studied linguistic phenomena

Studied linguistic phenomena

Generalized quantifiers

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 beings. 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 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".

Definite and indefinite 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 that the description can 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 danced" have the logical form $\exists x(Man(x)\land Danced(x))$. 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 a non-existent entity, posing challenges for determining its meaning and 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 $PDance(naomi)$ expresses that Naomi danced in the past, while $FDance(naomi)$ asserts that she will dance in the future. The semantic analysis of aspect is divided into morphological 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 $\exists eButter(Jones,thetoast,e)\land Slowly(e)\land With(e,aknife)$ (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

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 that it conveys. For example, the expressions "the morning star" and "the evening star" have the same extension since they both refer to the planet Venus. However, their meaning differs 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 one can substitute expressions with the same extension without changing the truth value of the sentence. For example, since the sentence "the morning star is a planet" is extensional, it 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 propositional attitudes 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 example above. The meaning of propositional attitude reports containing definite or indefinite descriptions is often ambiguous. The source of the ambiguity is the interpretation of the description, which can take a subjective or an objective perspective. For example, if Jasper wants a drink from his butler but is not aware 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. According to the subjective interpretation—called de dicto interpretation—it is false since Jasper does not want drinks from poisoners.

Questions and imperatives

The main focus of formal semantics has been 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 and specific frameworks have been proposed for the analysis of other types of sentences, such as questions and imperatives.

Various theories analyze the meaning of a question 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 questions "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 information that was asked for. 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 interogative sentences, which typically 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 for the analysis of imperatives are that imperative sentences usually lack an explicit subject and that they can be used to express various other meanings besides commands, such as advice, invitations, or permissions. Formal semanticists investigate 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, the operator $\lnot$ can be applied to statements to express negation, as in $\lnot Sleeps(mia)$ to state that Mia is not sleeping. This operator inverts the truth value of a statement, meaning that if the statement $Sleeps(mia)$ is false then the statement $\lnot Sleeps(mia)$ 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 only 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 either mean that not every doctor has a car, that not a single doctor has a car, or that there is no individual car owned by all doctors together.

Plural expressions are ways to 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 in a sentence applies to each individual separately or to the group as a whole. Some sentences are ambiguous in that both interpretations are possible. For example, the sentence "two boys pushed a car" can either mean that there were two cars and each boy pushed one (distributive) or that there was one car that the two 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 expressions. 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.

References

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