Atomic sentence
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In Logic, sentences (which are declarative sentences, also variously called propositions or statements) are those strings of words or symbols which are either true or false. In philosophy, such sentences are sometimes called truthbearers. The truth of some such sentences is a function of (is determined by) simpler sentences. (E.g. the truth of ‘John is Greek and John is Happy’ is a function of the truths of ‘John is Greek’ and ‘John is Happy’). The simplest kind of sentence, an elementary sentence, would not be a function of any logically simpler sentence, it is reasoned; other sentences built up from the elementary sentences using logical connectives such as 'and' and 'or' would be logically compound sentences, see Logical atomism.
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 (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. See vocabulary in First-order logic
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[edit] Examples
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 wfs are atomic sentences:
- p
- F(a)
- H(b,a,c)
The following wfs are atomic formulae but not atomic sentences because they include free variables:
- F(x)
- G(a,z)
- H(x,y,z)
The following wfs 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):
- F(x)&G(a,z)
- G(a,z)
H(x,y,z)
The following wfs are sentences but not atomic sentences (because they are not atomic formulae). (They are compound sentences):
x(F(x))
z(G(a,z))- xyz(H(x,y,x))
- xz(F(x)&G(a,z))
- xyz (G(a,z)H(x,y,z))
[edit] Interpretations
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:
Individual Constants
- a: Socrates
- b: Plato
- c: Aristotle
Predicates:
- Fα: α is sleeping
- Gαβ: α hates β
- Hαβγ: α made β hit γ
Sentential variables:
- 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."
[edit] 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
Predicates:
- 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.
[edit] Thoughts Quotes
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,[1]. A proposition (true or false) asserting an atomic fact is called an atomic proposition.: Russell, Introduction to Tractatus Logico-Philosophicus, [2]
Note the distinction between an elementary/atomic proposition and an atomic fact
The T-schema, which embodies the theory of truth proposed by Alfred Tarski, defines the truth of arbitrary sentences from the truth of atomic sentences.
[edit] See also
[edit] References
- 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: [3].]
