In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has additional meanings in mathematics. In a model, a function symbol will be modelled by a function.
Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F(X) is again a symbol representing an object in that language. 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.
Now consider a model of the formal language, with the types T and U modelled by sets [T] and [U] and each symbol X of type T modelled by an element [X] in [T]. Then F can be modelled by the set
which is simply a function with domain [T] and codomain [U]. It is a requirement of a consistent model that [F(X)] = [F(Y)] whenever [X] = [Y].
Introducing new function symbols
In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Given the function symbols F and G, one can introduce a new function symbol F ∘ G, the composition of F and G, satisfying (F ∘ G)(X) = F(G(X)), for allX. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain type of G, so this is required for the composition to be defined.
One also gets certain function symbols automatically. In untyped logic, there is an identity predicate id that satisfies id(X) = X for all X. In typed logic, given any type T, there is an identity predicate idT with domain and codomain type T; it satisfies idT(X) = X for all X of type T. Similarly, if T is a subtype of U, then there is an inclusion predicate of domain type T and codomain type U that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.
Additionally, one can define functional predicates after proving an appropriate theorem. (If you're working in a formal system that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.) Specifically, if you can prove that for every X (or every X of a certain type), there exists a uniqueY satisfying some condition P, then you can introduce a function symbol F to indicate this. Note that P will itself be a relational predicate involving both X and Y. So if there is such a predicate P and a theorem:
For all X of type T, for some unique Y of type U, P(X,Y),
then you can introduce a function symbol F of domain type T and codomain type U that satisfies:
For all X of type T, for all Y of type U, P(X,Y) if and only ifY = F(X).
Doing without functional predicates
Many treatments of predicate logic don't allow functional predicates, only relational predicates. This is useful, for example, in the context of proving metalogical theorems (such as Gödel's incompleteness theorems), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter). But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result.
Specifically, if F has domain type T and codomain type U, then it can be replaced with a predicate P of type (T,U). Intuitively, P(X,Y) means F(X) = Y. Then whenever F(X) would appear in a statement, you can replace it with a new symbol Y of type U and include another statement P(X,Y). To be able to make the same deductions, you need an additional proposition:
For allX of type T, for some uniqueY of type U, P(X,Y).
(Of course, this is the same proposition that had to be proven as a theorem before introducing a new function symbol in the previous section.)
Because the elimination of functional predicates is both convenient for some purposes and possible, many treatments of formal logic do not deal explicitly with function symbols but instead use only relation symbols; another way to think of this is that a functional predicate is a special kind of predicate, specifically one that satisfies the proposition above. This may seem to be a problem if you wish to specify a proposition schema that applies only to functional predicates F; how do you know ahead of time whether it satisfies that condition? To get an equivalent formulation of the schema, first replace anything of the form F(X) with a new variable Y. Then universally quantify over each Y immediately after the corresponding X is introduced (that is, after X is quantified over, or at the beginning of the statement if X is free), and guard the quantification with P(X,Y). Finally, make the entire statement a material consequence of the uniqueness condition for a functional predicate above.
First, we must replace F(C) with some other variable D:
Of course, this statement isn't correct; D must be quantified over just after C:
We still must introduce P to guard this quantification:
This is almost correct, but it applies to too many predicates; what we actually want is:
This version of the axiom schema of replacement is now suitable for use in a formal language that doesn't allow the introduction of new function symbols. Alternatively, one may interpret the original statement as a statement in such a formal language; it was merely an abbreviation for the statement produced at the end.
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.
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalisms that are no longer used and terminology that is often obscure. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem.
In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.
In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a canonical normal form, it is useful in automated theorem proving and circuit theory.
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". 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.
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
System F is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML. It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).
Predicate transformer semantics were introduced by Edsger Dijkstra in his seminal paper "Guarded commands, nondeterminacy and formal derivation of programs". They define the semantics of an imperative programming paradigm by assigning to each statement in this language a corresponding predicate transformer: a total function between two predicates on the state space of the statement. In this sense, predicate transformer semantics are a kind of denotational semantics. Actually, in guarded commands, Dijkstra uses only one kind of predicate transformer: the well-known weakest preconditions.
Independence-friendly logic is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form and , where is a finite set of variables. The intended reading of is "there is a which is functionally independent from the variables in ". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of IF sentences can characterize the same classes of structures as existential second-order logic. For example, it can express branching quantifier sentences, such as the formula which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in general, express this pattern of dependency, in which depends only on and , and depends only on and . IF logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix , which expresses that depends on , and depends on , but does not depend on .
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML.
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.
In logic, the monadic predicate calculus is the fragment of first-order logic in which all relation symbols in the signature are monadic, and there are no function symbols. All atomic formulas are thus of the form , where is a relation symbol and is a variable.
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.
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 is 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.
Bounded arithmetic is a collective name for a family of weak subtheories of Peano arithmetic. Such theories are typically obtained by requiring that quantifiers be bounded in the induction axiom or equivalent postulates. The main purpose is to characterize one or another class of computational complexity in the sense that a function is provably total if and only if it belongs to a given complexity class. Further, theories of bounded arithmetic present uniform counterparts to standard propositional proof systems such as Frege system and are, in particular, useful for constructing polynomial-size proofs in these systems. The characterization of standard complexity classes and correspondence to propositional proof systems allows to interpret theories of bounded arithmetic as formal systems capturing various levels of feasible reasoning.
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