- Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For example "The dog ran" is an atomic sentence in natural language, whereas "The dog ran and the cat hid." is a molecular sentence in natural language.
From a logical analysis, the truth or falsity of sentences in general is determined by only two things: the logical form of the sentence and the truth or falsity of its simple sentences. This is to say, for example, that the truth of the sentence "John is Greek and John is happy" is a function of the meaning of "and", and the truth values of the atomic sentences "John is Greek" and "John is happy". However, the truth or falsity of an atomic sentence is not a matter that is within the scope of logic itself, but rather whatever art or science the content of the atomic sentence happens to be talking about.
Logic has developed artificial languages, for example sentential calculus and predicate calculus partly with the purpose of revealing the underlying logic of natural languages statements, the surface grammar of which may conceal the underlying logical structure; see Analytic philosophy. In these artificial languages an Atomic Sentence is a string of symbols which can represent an elementary sentence in a natural language, and it can be defined as follows.
In a formal language, a well-formed formula (or wff) is a string of symbols constituted in accordance with the rules of syntax of the language. A term is a variable, an individual constant or a n-place function letter followed by n terms. An atomic formula is an wff consisting of either a sentential letter or an n-place predicate letter followed by n terms. A sentence is a wff in which any variables are bound. An atomic sentence is an atomic formula containing no variables. It follows that an atomic sentence contains no logical connectives, variables or quantifiers. A sentence consisting of one or more sentences and a logical connective is a compound (or molecular sentence). See vocabulary in First-order logic
As examples, let F, G, H be predicate letters; let a, b, c be individual constants; let x, y, z be variables; and let p be a sentential letter. Then the following wffs are atomic sentences:
The following wffs are atomic formulae but not atomic sentences because they include free variables:
The following wffs are not atomic formulae but are built up from atomic formulae using logical connectives. They are not sentences because they contain free variables. (They are compound formulae):
The following wffs are sentences but not atomic sentences (because they are not atomic formulae). (They are compound sentences):
- xyz (G(a,z)H(x,y,z))
A sentence is either true or false under an interpretation which assigns values to the logical variables. We might for example make the following assignments:
- a: Socrates
- b: Plato
- c: Aristotle
- Fα: α is sleeping
- Gαβ: α hates β
- Hαβγ: α made β hit γ
- p: It is raining.
Under this interpretation the sentences discussed above would represent the following English statements:
- p: "It is raining."
- F(a): "Socrates is sleeping."
- H(b,a,c): "Plato made Socrates hit Aristotle."
- x(F(x)): "Everybody is sleeping."
- z(G(a,z)): "Socrates hates somebody."
- xyz(H(x,y,z)): "Somebody made everybody hit somebody."
- xz(F(x)&G(a,z)): "Everybody is sleeping and Socrates hates somebody."
- xyz (G(a,z)H(x,y,z)): "Either Socrates hates somebody or somebody made everybody hit somebody."
Translating sentences from a natural language into an artificial language
Sentences in natural languages can be ambiguous, whereas the languages of the sentential logic and predicate logics are precise. Translation can reveal such ambiguities and express precisely the intended meaning.
For example take the English sentence "Father Ted married Jack and Jill". Does this mean Jack married Jill? In translating we might make the following assignments: Individual Constants
- a: Father Ted
- b: Jack
- c: Jill
- Mαβγ: α officiated at the marriage of β to γ
Using these assignments the sentence above could be translated as follows:
- M(a,b,c): Father Ted officiated at the marriage of Jack and Jill.
- xy((M(a,b,x)& (M(a,c,y)): ): Father Ted officiated at the marriage of Jack to somebody and Father Ted officiated at the marriage of Jill to somebody.
- xy(M(x,a,b)&M(y,a,c)): Somebody officiated at the marriage of Father Ted to Jack and somebody officiated at the marriage of Father Ted to Jill.
To establish which is the correct translation of "Father Ted married Jack and Jill", it would be necessary to ask the speaker exactly what was meant.
Atomic sentences are of particular interest in philosophical logic and the theory of truth and, it has been argued, there are corresponding atomic facts. An Atomic sentence (or possibly the meaning of an atomic sentence) is called an elementary proposition by Wittgenstein and an atomic proposition by Russell:
- 4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs. 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.: Wittgenstein, Tractatus Logico-Philosophicus, s:Tractatus Logico-Philosophicus.
- A proposition (true or false) asserting an atomic fact is called an atomic proposition.: Russell, Introduction to Tractatus Logico-Philosophicus, s:Tractatus Logico-Philosophicus/Introduction
- see also  and  especially regarding elementary proposition and atomic proposition as discussed by Russell and Wittgenstein
Note the distinction between an elementary/atomic proposition and an atomic fact
No atomic sentence can be deduced from (is not entailed by) any other atomic sentence, no two atomic sentences are incompatible, and no sets of atomic sentences are self-contradictory. Wittgenstein made much of this in his Tractatus Logico-Philosophicus. If there are any atomic sentences then there must be "atomic facts" which correspond to those that are true, and the conjunction of all true atomic sentences would say all that was the case, i.e. "the world" since, according to Wittegenstein, "The world is all that is the case". (TLP:1). Similarly the set of all sets of atomic sentences corresponds to the set of all possible worlds (all that could be the case).
- Benson Mates, Elementary Logic, OUP, New York 1972 (Library of Congress Catalog Card no.74-166004)
- Elliot Mendelson, Introduction to Mathematical Logic,, Van Nostran Reinholds Company, New York 1964
- Wittgenstein, Tractatus_Logico-Philosophicus: s:Tractatus Logico-Philosophicus.]
Wikimedia Foundation. 2010.
Look at other dictionaries:
atomic formula/sentence — Technically, an atomic formula is one with no logical constants . An atomic sentence is an atomic formula with no free variables . Intuitively, it is the kind of sentence on the basis of which one might start to construct logical complexity: a… … Philosophy dictionary
Atomic formula — In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms… … Wikipedia
Sentence (mathematical logic) — This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence, for a less technical introductory article see Statement (logic). In mathematical logic, a sentence of a… … Wikipedia
Atomic Spies — and Atom Spies are terms that refer to various people in the United States, Great Britain, and Canada who are thought to have illicitly given information about nuclear weapons production or design to the Soviet Union during World War II and the… … Wikipedia
atomic — formula/sentence … Philosophy dictionary
Debate over the atomic bombings of Hiroshima and Nagasaki — The Fat Man mushroom cloud resulting from the nuclear explosion over Nagasaki rises 18 km (11 mi, 60,000 ft) into the air from the hypocenter … Wikipedia
protocol sentence — ▪ philosophy in the philosophy of Logical Positivism, a statement that describes immediate experience or perception and as such is held to be the ultimate ground for knowledge. Such a statement is also called an atomic statement,… … Universalium
metalogic — /met euh loj ik/, n. the logical analysis of the fundamental concepts of logic. [1835 45; META + LOGIC] * * * Study of the syntax and the semantics of formal languages and formal systems. It is related to, but does not include, the formal… … Universalium
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 (a less… … Wikipedia
Semantic theory of truth — A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences. Contents 1 Origin 2 Tarski s Theory 3 See also … Wikipedia