Interpretation (logic)An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.
First-order logicFirst-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.
Parallel textA parallel text is a text placed alongside its translation or translations. Parallel text alignment is the identification of the corresponding sentences in both halves of the parallel text. The Loeb Classical Library and the Clay Sanskrit Library are two examples of dual-language series of texts. Reference Bibles may contain the original languages and a translation, or several translations by themselves, for ease of comparison and study; Origen's Hexapla (Greek for "sixfold") placed six versions of the Old Testament side by side.
Semantic similaritySemantic similarity is a metric defined over a set of documents or terms, where the idea of distance between items is based on the likeness of their meaning or semantic content as opposed to lexicographical similarity. These are mathematical tools used to estimate the strength of the semantic relationship between units of language, concepts or instances, through a numerical description obtained according to the comparison of information supporting their meaning or describing their nature.
Lexical semanticsLexical semantics (also known as lexicosemantics), as a subfield of linguistic semantics, is the study of word meanings. It includes the study of how words structure their meaning, how they act in grammar and compositionality, and the relationships between the distinct senses and uses of a word. The units of analysis in lexical semantics are lexical units which include not only words but also sub-words or sub-units such as affixes and even compound words and phrases. Lexical units include the catalogue of words in a language, the lexicon.
Atomic formulaIn mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
Word-sense disambiguationWord-sense disambiguation (WSD) is the process of identifying which sense of a word is meant in a sentence or other segment of context. In human language processing and cognition, it is usually subconscious/automatic but can often come to conscious attention when ambiguity impairs clarity of communication, given the pervasive polysemy in natural language. In computational linguistics, it is an open problem that affects other computer-related writing, such as discourse, improving relevance of search engines, anaphora resolution, coherence, and inference.
Universal quantificationIn mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Non-logical symbolIn logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants). The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates.
