A finite-state machine (FSM) or finite-state automaton, finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of states at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a transition. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—deterministic finite-state machines and non-deterministic finite-state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
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.
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In linguistics, syntax is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning (semantics). There are numerous approaches to syntax that differ in their central assumptions and goals.
Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".
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 logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic.
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.
Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz and in the 1950s by Yehoshua Bar-Hillel and Joachim Lambek. It saw a surge of interest in the 1970s following the work of Richard Montague, whose Montague grammar assumed a similar view of syntax. It continues to be a major paradigm, particularly within formal semantics.
Meta is a prefix meaning "more comprehensive" or "transcending".
The term predicate is used in one of two ways in linguistics and its subfields. The first defines a predicate as everything in a standard declarative sentence except the subject, and the other views it as just the main content verb or associated predicative expression of a clause. Thus, by the first definition the predicate of the sentence Frank likes cake is likes cake. By the second definition, the predicate of the same sentence is just the content verb likes, whereby Frank and cake are the arguments of this predicate. Differences between these two definitions can lead to confusion.
A metatheory or meta-theory is a theory whose subject matter is theory itself, aiming to describe existing theory in a systematic way. In mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. Meta-theoretical investigations are part of the philosophy of science. A metatheory is not applied directly to practice, but may have applications to the practice of the field it studies. The emerging field of metascience seeks to use scientific knowledge to improve the practice of science itself.
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In mathematical logic, a judgment or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.
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.
In logic, a metavariable is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence
