This article may be too technical for most readers to understand.(March 2016) |
In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic which is true in a model if and only if the statement holds in that model. Nonfirstorderizable statements are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural language.
The term was coined by George Boolos in his paper "To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)". [1] Quine argued that such sentences call for second-order symbolization, which can be interpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct "second-order objects" (properties, sets, etc.).
A standard example is the Geach–Kaplan sentence: "Some critics admire only one another." If Axy is understood to mean "x admires y," and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is: In words, this states that there exists a collection of critics with the following properties: The collection forms a proper subclass of all the critics; it is inhabited (and thus non-empty) by a member that admires a critic that is also a member; and it is such that if any of its members admires anyone, then the latter is necessarily also a member.
That this formula has no first-order equivalent can be seen by turning it into a formula in the language of arithmetic. To this end, substitute the formula for Axy. This expresses that the two terms are successors of one another, in some way. The resulting proposition, states that there is a set X with the following three properties:
Recall a model of a formal theory of arithmetic, such as first-order Peano arithmetic, is called standard if it only contains the familiar natural numbers as elements (i.e., 0, 1, 2, ...). The model is called non-standard otherwise. The formula above is true only in non-standard models: In the standard model X would be a proper subset of all numbers that also would have to contain all available numbers (0, 1, 2, ...), and so it fails. And then on the other hand, in every non-standard model there is a subset X satisfying the formula.
Let us now assume that there is a first-order rendering of the above formula called E. If were added to the Peano axioms, it would mean that there were no non-standard models of the augmented axioms. However, the usual argument for the existence of non-standard models would still go through, proving that there are non-standard models after all. This is a contradiction, so we can conclude that no such formula E exists in first-order logic.
There is no formula A in first-order logic with equality which is true of all and only models with finite domains. In other words, there is no first-order formula which can express "there is only a finite number of things".
This is implied by the compactness theorem as follows. [2] Suppose there is a formula A which is true in all and only models with finite domains. We can express, for any positive integer n, the sentence "there are at least n elements in the domain". For a given n, call the formula expressing that there are at least n elements Bn. For example, the formula B3 is: which expresses that there are at least three distinct elements in the domain. Consider the infinite set of formulae Every finite subset of these formulae has a model: given a subset, find the greatest n for which the formula Bn is in the subset. Then a model with a domain containing n elements will satisfy A (because the domain is finite) and all the B formulae in the subset. Applying the compactness theorem, the entire infinite set must also have a model. Because of what we assumed about A, the model must be finite. However, this model cannot be finite, because if the model has only m elements, it does not satisfy the formula Bm+1. This contradiction shows that there can be no formula A with the property we assumed.
First-order logic—also called predicate logic, predicate calculus, quantificational logic—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. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x, if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "... is a man" and "... is mortal" are predicates. 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.
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 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", "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.
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 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.
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic, it provides a canonical normal form useful in automated theorem proving.
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.
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe.
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity. As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" and "congruence". The system contains infinitely many axioms.
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction.
In model theory and related areas of mathematics, a type is an object that describes how a element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,..., xn that are true of a set of n-tuples of an L-structure . Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure . The question of which types represent actual elements of leads to the ideas of saturated models and omitting types.
Axiomatic 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.
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.
In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula is satisfiable because it is true when and , while the formula is not satisfiable over the integers. The dual concept to satisfiability is validity; a formula is valid if every assignment of values to its variables makes the formula true. For example, is valid over the integers, but is not.
In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all.
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 mathematical logic, the spectrum of a sentence is the set of natural numbers occurring as the size of a finite model in which a given sentence is true. By a result in descriptive complexity, a set of natural numbers is a spectrum if and only if it can be recognized in non-deterministic exponential time.