In logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, the term decidable refers to the existence of an effective method Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability for determining membership in a set of formulas. Logical systems In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its such as propositional logic In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory In mathematical logic, a theory is a set of sentences in a formal language. For example, a first-order theory is a set of first-order sentences. Many authors require that the theory be closed under logical consequence (set of formulas closed under logical consequence Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory&) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory.
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Relationship to computability
As with the concept of a decidable set In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. A set which is not computable is called noncomputable or undecidable, the definition of a decidable theory or logical system can be given either in terms of effective methods Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability or in terms of computable functions Computable functions are the basic objects of study in computability theory. The set of computable functions is equivalent to the set of Turing-computable functions and partial recursive functions. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any. These are generally considered equivalent per Church's thesis In computability theory the Church–Turing thesis is a combined hypothesis ("thesis") about the nature of effectively calculable (computable) functions by recursion (Church's Thesis), by mechanical device equivalent to a Turing machine (Turing's Thesis) or by use of Church's λ-calculus. The three computational processes (recursion, λ-. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp. 206ff.).
Decidability of a logical system
Each logical system In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its comes with both a syntactic component 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, which among other things determines the notion of provability A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a, and a semantic component Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas, which determines the notion of logical validity. The logically valid formulas of a system are sometimes called the theorems of the system, especially in the context of first-order logic where Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929 establishes the equivalence of semantic and syntactic consequence. In other settings, such as linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such, the syntactic consequence (provability) relation may be used to define the theorems of a system.
A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, propositional logic In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed is decidable, because the truth-table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables . In particular, truth tables method can be used to determine whether an arbitrary propositional formula In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula is logically valid.
First-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each is not decidable in general; in particular, the set of logical validities in any signature In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes that includes equality and at least one other predicate with two or more arguments is not decidable. Logical systems extending first-order logic, such as second-order logic 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 and type theory In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general. In programming language theory, a branch of computer science, type theory can refer to the design, analysis and study of type systems, although some computer, are also undecidable.
The validities of monadic predicate calculus In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic , and there are no function letters. All atomic formulae have the form P(x), where P is a predicate letter and x is a variable with identity are decidable, however. This system is first-order logic restricted to signatures that have no function symbols and whose relation symbols other than equality never take more than one argument.
Some logical systems are not adequately represented by the set of theorems alone. (For example, Kleene's logic has no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity of sequents In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system, or the consequence relation Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory& {(Г, A) | Г ⊧ A} of the logic.
Decidability of a theory
A theory In mathematical logic, a theory is a set of sentences in a formal language. For example, a first-order theory is a set of first-order sentences. Many authors require that the theory be closed under logical consequence is a set of formulas, which here is assumed to be closed under logical consequence Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory&. The question of decidability for a theory is whether there is an effective procedure that, given an arbitrary formula in the signature of the theory, decides whether the formula is a member of the theory or not. This problem arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms. Examples of decidable first-order theories include the theory of real closed fields In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers, and Presburger arithmetic, while the theory of groups In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from and Robinson arithmetic are examples of undecidable theories.
There are several basic results about decidability of theories. Every inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus member of, the theory. Every complete In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this is equivalent to asking that for every sentence φ in the language of the theory it contains either φ recursively enumerable In computational complexity theory, the complexity class containing all recursively enumerable sets is RE. In recursion theory, the lattice of r.e. sets under inclusion is denoted first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.
A consistent theory which has the property that every consistent extension is undecidable is said to be essentially undecidable. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable. Robinson arithmetic is known to be essentially undecidable, and thus every consistent theory which includes or interprets Robinson arithmetic is also (essentially) undecidable.
Some decidable theories
Some decidable theories include (Monk 1976, p. 234):
- The set of first-order logical validities in the signature with only equality, established by Leopold Löwenheim in 1915.
- The set of first-order logical validities in a signature with equality and one unary function, established by Ehrenfeucht in 1959.
- The first-order theory of the integers in the signature with equality and addition, also called Presburger arithmetic. The completeness was established by Mojżesz Presburger in 1929.
- The first-order theory of Boolean algebras, established by Alfred Tarski Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1949.
- The first-order theory of algebraically closed fields of a given characteristic, established by Tarski in 1949.
- The first-order theory of real-closed ordered fields, established by Tarski in 1949.
- The first-order theory of Euclidean geometry, established by Tarski in 1949.
- The first-order theory of hyperbolic geometry, established by Schwabhäuser in 1959.
- Specific decidable sublanguages of set theory investigated in the 1980s through today.(Cantone et al., 2001)
Methods used to establish decidability include quantifier elimination Quantifier elimination is a concept in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification. Formulae with less depth of quantifier alternation are thought of as simpler and the quantifier free formulae as the simplest. A theory has quantifier elimination if for, model completeness, and Vaught's test.
Some undecidable theories
Some undecidable theories include (Monk 1976, p. 279):
- The set of logical validities in any first-order signature with equality and either: a relation symbol of arity no less than 2, or two unary function symbols, or one function symbol of arity no less than 2, established by Trakhtenbrot in 1953.
- The first-order theory of the natural numbers with addition, multiplication, and equality, established by Tarski and Andrzej Mostowski in 1949.
- The first-order theory of the rational numbers with addition, multiplication, and equality, established by Julia Robinson in 1949.
- The first-order theory of groups, established by Mal'cev in 1961. Mal'cev also established that the theory of semigroups and the theory of rings are undecidable. Robinson established in 1949 that the theory of fields is undecidable.
- Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of 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 is essentially undecidable. Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that show that many other sufficiently strong theories of arithmetic share this property.
The interpretability Assume T and S are formal theories. Slightly simplified, T is said to be interpretable in S if and only if the language of T can be translated into the language of S in such a way that S proves the translation of every theorem of T. Of course, there are some natural conditions on admissible translations here, such as the necessity for a method is often used to establish undecidability of theories. If an essentially undecidable theory T is interpretable in a consistent theory S, then S is also essentially undecidable. This is closely related to the concept of a many-one reduction in computability theory.
Semidecidability
A property of a theory or logical system weaker than decidability is semidecidability. A theory is semidecidable if there is an effective method which, given an arbitrary formula, will always tell correctly when the formula is in the theory, but may give either a negative answer or no answer at all when the formula is not in the theory. A logical system is semidecidable if there is an effective method for generating theorems (and only theorems) such that every theorem will eventually be generated. This is different from decidability because in a semidecidable system there may be no effective procedure for checking that a formula is not a theorem.
Every decidable theory or logical system is semidecidable, but in general the converse is not true; a theory is decidable if and only if both it and its complement are semi-decidable. For example, the set of logical validities V of first-order logic is semi-decidable, but not decidable. In this case, it is because there is no effective method for determining for an arbitrary formula A whether A is not in V. Similarly, the set of logical consequences of any recursively enumerable set In computational complexity theory, the complexity class containing all recursively enumerable sets is RE. In recursion theory, the lattice of r.e. sets under inclusion is denoted of first-order axioms is semidecidable. Many of the examples of undecidable first-order theories given above are of this form.
Relationship with completeness
Decidability should not be confused with completeness In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this is equivalent to asking that for every sentence φ in the language of the theory it contains either φ. For example, the theory of algebraically closed fields In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable. Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym for independent statement In mathematical logic, independence refers to the unprovability of a sentence from other sentences.
See also
- László Kalmár (1936)
- Alonzo Church Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem (1956)
- W.V.O. Quine Willard Van Orman Quine (known to intimates as "Van") was an American philosopher and logician in the analytic tradition. From 1930 until his death 70 years later, Quine was continuously affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of mathematics, and (1953)
- Meyer and Lambert (1967)
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logic For example you can adapt what I called the Second Order Liar to a Second Order Goedel Sentence one whose decidability is undecidable Given an arithmetical sentence let be the sentence asserting that is provable in PA Peano Arithmetic Then by diagonalization or fixed point or some such proposition we know that there is a sentence such that Peano
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arxiv_cs
Fri, 05 Sep 2008 05:42:04 GM
We also present some (un). decidability. results that are related to the above separation properties for AL: the undecidability of $_=L$ on MA and its . decidability. on the subcalculus. read more at cs updates on arXiv.org. rss2lj.
