To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Sentence (mathematical logic)

From Wikipedia, the free encyclopedia

This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence (linguistics), for a less technical introductory article 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.

YouTube Encyclopedic

  • 1/3
    Views:
    81 347
    20 687
    290 356
  • Part 1: Symbolic Logic (The basics, letters, operators, connectives)
  • [Discrete Math 1] Statement Identification and Translation Examples
  • Truth Table Tutorial - Discrete Mathematics Logic

Transcription

Example

The following example in first-order logic

is a sentence. This sentence is true in the positive real numbers+, false in the real numbers ℝ, and true in the complex numbers ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the square of a member of that particular structure.) On the other hand, the formula

is not a sentence, because of the presence of the free variable y. In the structure of the real numbers, this formula is true if we substitute (arbitrarily) y = 2, but is false if y = –2.

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

See also

References

  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.
This page was last edited on 29 December 2021, at 19:45
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.