Concept
Sentence (mathematical logic)
Overview
In mathematical logic, a sentence (or closed formula) 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 sentences 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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Sentence (mathematical logic) is discussed.
If untenable in finite psychology, that opinion ought to be untenable in metaphysics also. The great transcendentalist metaphor has always been, as I lately reminded you, a grammatical sentence. Physically such a sentence is of course composed of clauses, these of words, the words of syllables, and the syllables of letters. We may take each word in, yet not understand the sentence; but if suddenly the meaning of the…
William James, A Pluralistic Universe
names the idea; states what it is
It is something altogether more abstract and remote. We might understand all the separate words of a sentence without understanding the sentence: if a sentence is long and complicated, this is apt to happen. In such a case we have knowledge of the constituents, but not of the form. We may also have knowledge of the form without having knowledge of the constituents.
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
names the idea; states what it is
For every sentence is previously formed and circumscribed in the mind of the Speaker, and is then immediately attended by the proper words to express it, which the same mental faculty (than which there is nothing more lively and expeditious) instantly dismisses, and sends off each to its proper post: but, in different sentences, their particular order and arrangement will be differently terminated; though, in every…
Marcus Tullius Cicero, Cicero's Brutus or History of Famous Orators
names the idea; states what it is; opens on it
The grammatical meaning of these categories shows clearly that this analysis consists of a system of primary and secondary relations, which are joined into a unified resultant by this logical arrangement. Thus the relation of subject and predicate includes all those further relations of noun and adjective, verb and object or adverb, as its minor terms, which are joined together partly by their own relations and…
Wilhelm Max Wundt, An Introduction to Psychology Translated from the Second German Edition
states what it is; names the idea; opens on it
For the sentence is first composed in the mind; and then words come immediately: and then they are immediately sent forth by the mind, than which nothing is more rapid in its movements; so that each falls into its proper place. And then their regular order is settled by different terminations in different sentences; and all the expressions at the beginning and in the middle of the sentence ought to be composed with…
Marcus Tullius Cicero, The Orations of Marcus Tullius Cicero, Volume 4
names the idea; states what it is
Related ideas
- First-order logic
- Free variables and bound variables
- Truth value
- Logical connective
- Mathematical logic
- Domain of discourse
- Theorem
- Well-formed formula
- Structure (mathematical logic)
- Quantifier (logic)
- Atomic formula
- Boolean algebra
- Complex number
- Satisfiability
Place in the tradition
Part of Outline of logic.
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