In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and
analytic philosophy
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...
, an atomic sentence is a type of declarative
sentence which is either true or false (may also be referred to as a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
,
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
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
.
From a logical analysis point of view, 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
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boole ...
", and the
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false'').
Computing
In some progr ...
s 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-language statements, the surface
grammar
In linguistics, the grammar of a natural language is its set of structure, structural constraints on speakers' or writers' composition of clause (linguistics), clauses, phrases, and words. The term can also refer to the study of such constraint ...
of which may conceal the underlying logical structure. 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
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...
(or wff) is a string of symbols constituted in accordance with the rules of syntax of the language. A
term is a
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
, an individual
constant or a n-place
function letter In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and ''Arity, n-ary'' form. Function symbols are used, together with constants and variables, to form term (logic), terms.
The theory ...
followed by n terms. An
atomic formula is a wff consisting of either a sentential letter or an n-place
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
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
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s, variables or
quantifiers. A sentence consisting of one or more sentences and a logical connective is a compound (or molecular) sentence.
Examples
Assumptions
In the following examples:
* let ''F'', ''G'', ''H'' be predicate letters;
* let ''a'', ''b'', ''c'' be individual constants;
* let ''x'', ''y'', ''z'' be variables.
Atomic sentences
These wffs are atomic sentences; they contain no free variables or conjunctions:
* ''F''(''a'')
* ''G''(''a'', ''b'')
* ''H''(''a'', ''b'', ''c'')
Atomic formulae
These wffs are atomic formulae, but are not sentences (atomic or otherwise) because they include free variables:
* ''F''(''x'')
* ''G''(''a'', ''z'')
* ''H''(''x'', ''y'', ''z'')
Compound sentences
These wffs are compound sentences. They are sentences, but are not atomic sentences because they are not atomic formulae:
* ∀''x'' (''F''(''x''))
* ∃''z'' (''G''(''a'', ''z''))
* ∃''x'' ∀''y'' ∃''z'' (''H''(''x'', ''y'', ''z''))
* ∀''x'' ∃''z'' (''F''(''x'') ∧ ''G''(''a'', ''z''))
* ∃''x'' ∀''y'' ∃''z'' (''G''(''a'', ''z'') ∨ ''H''(''x'', ''y'', ''z''))
Compound formulae
These wffs are compound formulae. They are not atomic formulae but are built up from atomic formulae using logical connectives. They are also not sentences because they contain free variables:
* ''F''(''x'') ∧ ''G''(''a'', ''z'')
* ''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."
* ∃''x'' ∀''y'' ∃''z'' (''H''(''x'', ''y'', ''z'')): "Somebody made everybody hit somebody." (They may not have all hit the same person z, but they all did so ''because'' of the same person x.)
* ∀''x'' ∃''z'' (''F''(''x'') ∧ ''G''(''a'', ''z'')): "Everybody is sleeping and Socrates hates somebody."
* ∃''x'' ∀''y'' ∃''z'' (''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
Predicates:
* ''M''αβγ: 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.
* ∃''x'' ∃''y'' (''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.
* ∃''x'' ∃''y'' (''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 logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
and the theory of
truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
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
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considere ...
and an atomic proposition by
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
:
* ''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
The ''Tractatus Logico-Philosophicus'' (widely abbreviated and cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to define the ...
''.
* ''A proposition (true or false) asserting an atomic fact is called an atomic proposition.'' — Russell, "Introduction to ''Tractatus Logico-Philosophicus''"
* 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''. 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.
See also
*
Logical constant
References
Bibliography
* Benson Mates, ''Elementary Logic'', Oxford University Press, 1972.
* Elliot Mendelson, ''Introduction to Mathematical Logic'', Van Nostran Reinholds Company, 1964.
{{Mathematical logic
Predicate logic
Sentences by type
Syntax (logic)
Propositions