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First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions. For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes. For instance if one believes that the sky is blue, what one believes is the proposition that the sky is blue. A proposition can also be thought of as a kind of idea: Collins Dictionary has a definition for proposition as "a statement or an idea that people can consider or discuss whether it is true."
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language.
In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 stories.
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
Irene Roswitha Heim is a linguist and a leading specialist in semantics. She was a professor at the University of Texas at Austin and UCLA before moving to the Massachusetts Institute of Technology in 1989, where she is Professor Emerita of Linguistics. She served as Head of the Linguistics Section of the Department of Linguistics and Philosophy.
David Benjamin Kaplan is an American philosopher. He is the Hans Reichenbach Professor of Scientific Philosophy at the UCLA Department of Philosophy. His philosophical work focuses on the philosophy of language, logic, metaphysics, epistemology and the philosophy of Frege and Russell. He is best known for his work on demonstratives, propositions, and reference in intensional contexts. He was elected a Fellow of the American Academy of Arts & Sciences in 1983 and a Corresponding Fellow of the British Academy in 2007.
In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and J. Michael Dunn and Nuel Belnap. It is also called the substitution interpretation or substitutional quantification.
In formal linguistics, discourse representation theory (DRT) is a framework for exploring meaning under a formal semantics approach. One of the main differences between DRT-style approaches and traditional Montagovian approaches is that DRT includes a level of abstract mental representations within its formalism, which gives it an intrinsic ability to handle meaning across sentence boundaries. DRT was created by Hans Kamp in 1981. A very similar theory was developed independently by Irene Heim in 1982, under the name of File Change Semantics (FCS). Discourse representation theories have been used to implement semantic parsers and natural language understanding systems.
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
Donkey sentences are sentences that contain a pronoun with clear meaning but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse such sentences.
Quantificational variability effect (QVE) is the intuitive equivalence of certain sentences with quantificational adverbs (Q-adverbs) and sentences without these, but with quantificational determiner phrases (DP) in argument position instead.
Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it brings about in the information state of anyone who accepts the news conveyed by it." In dynamic semantics, sentences are mapped to functions called context change potentials, which take an input context and return an output context. Dynamic semantics was originally developed by Irene Heim and Hans Kamp in 1981 to model anaphora, but has since been applied widely to phenomena including presupposition, plurals, questions, discourse relations, and modality.
Formal semantics is the study of grammatical meaning in natural languages using formal tools 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.
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
In formal semantics, the scope of a semantic operator is the semantic object to which it applies. For instance, in the sentence "Paulina doesn't drink beer but she does drink wine," the proposition that Paulina drinks beer occurs within the scope of negation, but the proposition that Paulina drinks wine does not. Scope can be thought of as the semantic order of operations.
In formal semantics, a type shifter is an interpretation rule that changes an expression's semantic type. For instance, the English expression "John" might ordinarily denote John himself, but a type shifting rule called Lift can raise its denotation to a function which takes a property and returns "true" if John himself has that property. Lift can be seen as mapping an individual onto the principal ultrafilter that it generates.
- Without type shifting:
- Type shifting with Lift:
In linguistics, the syntax–semantics interface is the interaction between syntax and semantics. Its study encompasses phenomena that pertain to both syntax and semantics, with the goal of explaining correlations between form and meaning. Specific topics include scope, binding, and lexical semantic properties such as verbal aspect and nominal individuation, semantic macroroles, and unaccusativity.
An indeterminate pronoun is a pronoun which can show a variety of readings depending on the type of sentence it occurs in. The term "indeterminate pronoun" originates in Kuroda's (1965) thesis and is typically used in reference to wh-indeterminates, which are pronouns which function as an interrogative pronoun in questions, yet come to have additional meanings with other grammatical operators. For example, in Japanese, dare means 'who' in a constituent question like (1) formed with the question-forming operator no:
References
- ↑ Brasoveanu, Adrian; Farkas, Donka (2016). "Indefinites". In Aloni, Maria; Dekker, Paul (eds.). The Cambridge Handbook of Formal Semantics. Cambridge University Press. pp. 238–266. doi:10.1017/CBO9781139236157.009. ISBN 9781107028395.
- ↑ Heim, Irene (1982). "Chapter 2: Indefinites as Variables". The Semantics of Definite and Indefinite Noun Phrases (PDF) (Thesis). University of Massachusetts, Amherst.
- ↑ Kratzer, Angelika; Shimoyama, Junko (2002). "Indeterminate pronouns: The view from Japanese" (PDF). Proceedings of the Third Tokyo Conference on Psycholinguistics.
- ↑ Ciardelli, Ivano; Roelofsen, Floris; Theiler, Nadine (2017). "Composing alternatives" (PDF). Linguistics and Philosophy. 40 (1): 1–36. doi: 10.1007/s10988-016-9195-2 .
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