Atomic sentence
Encyclopedia
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 connective
s, 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
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):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))
Individual Constants
Predicates:
Sentential variables:
Under this interpretation the sentences discussed above would represent the following English statements:
s 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
Predicates:
Using these assignments the sentence above could be translated as follows:
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.
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:
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).
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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, an atomic sentence is a type of declarative sentence
Sentence (mathematical logic)
In mathematical logic, a sentence 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 may be true or false...
which is either true or false (may also be referred to as a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
, statement
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...
or truthbearer
Truthbearer
Truth-bearer is a term used to designate entities that are either true or false and nothing else. The thesis that some things are true while others are false raises the question of the nature of these things. Since there is divergence of opinion on the matter, the term truthbearer is used to be...
) 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
Logical form
In logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a 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
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
", 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
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
(or wff) is a string of symbols constituted in accordance with the rules of syntax of the language. A term
Term (logic)
In mathematical logic, universal algebra, and rewriting systems, terms are expressions which can be obtained from constant symbols, variables and function symbols...
is a variable, an individual constant or a n-place function letter followed by n terms. An atomic formula
Atomic formula
In mathematical logic, an atomic formula 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 are thus the simplest well-formed formulas of the logic...
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 connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s, 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
First-order logic
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...
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 wffs are atomic sentences:- p
- F(a)
- H(b,a,c)
The following wffs are atomic formulae but not atomic sentences because they include free variables:
- F(x)
- G(a,z)
- H(x,y,z)
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):
- F(x)&G(a,z)
- G(a,z)H(x,y,z)
The following wffs 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))
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."
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 logicPredicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
s 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.
Philosophical significance
Atomic sentences are of particular interest in philosophical logicPhilosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
and the theory of truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
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-PhilosophicusTractatus Logico-PhilosophicusThe Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. It was an ambitious project: to identify the relationship between language and reality and to define the limits of science...
, 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).
The T-schema
T-schema
The T-schema or truth schema is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth...
, which embodies the theory of truth proposed by Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, defines the truth of arbitrary sentences from the truth of atomic sentences.