sentence (mathematical logic)

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Sentence (linguistics) In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include p ...
, for a less technical introductory article see
Statement (logic)In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
.'' In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal syst ...
, a sentence (or closed formula)Edgar Morscher, "Logical Truth and Logical Form", ''Grazer Philosophische Studien'' 82(1), pp. 77–90. of a
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of ...
is a Boolean-valued
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 symbol (formal), symbols from a given alphabet (computer science), alphabet that is part of ...
with no
free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expressio ...
. A sentence can be viewed as expressing a
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...
, 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 value In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
s: 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 connective In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
s or
quantifiers Quantifier may refer to: * Quantifier (linguistics), an indicator of quantity * Quantifier (logic) * Quantification (science) See also

*Quantification (disambiguation) {{disambiguation ...
in them are known as
atomic sentenceIn logic and analytic philosophy, an atomic sentence is a type of declarative Sentence (mathematical logic), sentence which is either true or false (may also be referred to as a proposition, statement (logic), statement or truthbearer) and which cann ...
s; by analogy to atomic formula. Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a Theory (mathematical logic), 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 (logic), interpretation of the theory. For first-order theories, interpretations are commonly called structure (mathematical logic), structures. Given a structure or interpretation, a sentence will have a fixed
truth value In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
. A theory is Satisfiability, 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.

# Example

The following example in first-order logic :$\forall y\exists x \left(x^2=y\right)$ a sentence. This sentence is true in the positive real numbers ℝ+, false in the real numbers ℝ, and true in the complex numbers ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned is the Square (algebra), square of a member of that particular structure.) On the other hand, the formula :$\exists x\left(x^2=y\right)$ is a sentence, because of the presence of the free variable ''y''. In the structure of the real numbers, this formula is true if we substitute (arbitrarily) ''y'' = 2, but is false if ''y'' = –2. It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even in the structure of the complex numbers, where the statement is always true, it is still not considered a sentence. Such a formula may be called a predicate (logic), predicate instead.