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Sentence (mathematical logic)

In mathematical logic, a sentence (or closed formula)[1] of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.

Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic formulas by applying connectives and quantifiers.

A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.

Example edit

For the interpretation of formulas, consider these structures: the positive real numbers, the real numbers, and complex numbers. The following example in first-order logic

 

is a sentence. This sentence means that for every y, there is an x such that   This sentence is true for positive real numbers, false for real numbers, and true for complex numbers.

However, the formula

 

is not a sentence because of the presence of the free variable y. For real numbers, this formula is true if we substitute (arbitrarily)   but is false if  

It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even for complex numbers, where the formula is always true, it is still not considered a sentence. Such a formula may be called a predicate instead.

See also edit

References edit

  1. ^ Edgar Morscher, "Logical Truth and Logical Form", Grazer Philosophische Studien 82(1), pp. 77–90.
  • Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.
  • Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.

sentence, mathematical, logic, more, accessible, less, technical, introduction, this, topic, statement, logic, mathematical, logic, sentence, closed, formula, predicate, logic, boolean, valued, well, formed, formula, with, free, variables, sentence, viewed, ex. For a more accessible and less technical introduction to this topic see Statement logic In mathematical logic a sentence or closed formula 1 of a predicate logic is a Boolean valued well formed formula with no free variables A sentence can be viewed as expressing a proposition something that must be true or false The restriction of having no free variables is needed to make sure that sentences can have concrete fixed truth values as the free variables of a general formula can range over several values the truth value of such a formula may vary Sentences without any logical connectives or quantifiers in them are known as atomic sentences by analogy to atomic formula Sentences are then built up out of atomic formulas by applying connectives and quantifiers A set of sentences is called a theory thus individual sentences may be called theorems To properly evaluate the truth or falsehood of a sentence one must make reference to an interpretation of the theory For first order theories interpretations are commonly called structures Given a structure or interpretation a sentence will have a fixed truth value A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem Example editFor the interpretation of formulas consider these structures the positive real numbers the real numbers and complex numbers The following example in first order logic y x y x2 displaystyle forall y exists x y x 2 nbsp is a sentence This sentence means that for every y there is an x such that y x2 textstyle y x 2 nbsp This sentence is true for positive real numbers false for real numbers and true for complex numbers However the formula x y x2 displaystyle exists x y x 2 nbsp is not a sentence because of the presence of the free variable y For real numbers this formula is true if we substitute arbitrarily y 2 textstyle y 2 nbsp but is false if y 2 textstyle y 2 nbsp It is the presence of a free variable rather than the inconstant truth value that is important for example even for complex numbers where the formula is always true it is still not considered a sentence Such a formula may be called a predicate instead See also editGround expression Open formula Statement logic PropositionReferences edit Edgar Morscher Logical Truth and Logical Form Grazer Philosophische Studien 82 1 pp 77 90 Hinman P 2005 Fundamentals of Mathematical Logic A K Peters ISBN 1 56881 262 0 Rautenberg Wolfgang 2010 A Concise Introduction to Mathematical Logic 3rd ed New York Springer Science Business Media doi 10 1007 978 1 4419 1221 3 ISBN 978 1 4419 1220 6 nbsp This logic related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Sentence mathematical logic amp oldid 1160900785, wikipedia, wiki, book, books, library,

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