A formal system (or deductive system) is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.However, in 1931 Kurt Gödel proved that any consistent formal system sufficiently powerful to express basic arithmetic cannot prove its own completeness. This effectively showed that Hilbert's program was impossible as stated.
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, game theory, systems theory, decision theory and theoretical linguistics. Whereas the natural sciences and social sciences seek to characterize physical systems and social systems, respectively, using theoretical and empirical methods, the formal sciences use language tools concerned with characterizing abstract structures described by formal systems and the deductions that can be made from them. The formal sciences aid the natural and social sciences by providing information about the structures used to describe the physical world, and what inferences may be made about them.
Formal semantics is the scientific study of linguistic meaning through formal tools from logic and mathematics. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. Formal semanticists rely on diverse methods to analyze natural language. Many examine the meaning of a sentence by studying the circumstances in which it would be true. They describe these circumstances using abstract mathematical models to represent entities and their features. The principle of compositionality helps them link the meaning of expressions to abstract objects in these models. This principle asserts that the meaning of a compound expression is determined by the meanings of its parts.
Propositional and predicate logic are formal systems used to analyze the semantic structure of sentences. They introduce concepts like singular terms, predicates, quantifiers, and logical connectives to represent the logical form of natural language expressions. Type theory is another approach utilized to describe sentences as nested functions with precisely defined input and output types. Various theoretical frameworks build on these systems. Possible world semantics and situation semantics evaluate truth across different hypothetical scenarios. Dynamic semantics analyzes the meaning of a sentence as the information contribution it makes.
In formal systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though, formally, does not need to represent anything at all. Function symbols are a basic component in formal languages to form terms. Specifically, if the symbol is a function symbol, then given any constant symbol representing an object in the language, also represents an object in the language. Similarly, if is some term in the language, is also a term. As such, the interpretation of a function symbol must be defined over the whole domain of discourse. Function symbols are a primitive notion, and are therefore not defined in terms of other, more basic concepts.
In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U.One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol.
