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 ...

, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

and

higher-order logics mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...

Uses

Formulas in logic are typically built up recursively from some propositional variables, some number of

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 ...

s, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as

P

Q

and

R

. ;Example In a given propositional logic, a formula can be defined as follows: * Every propositional variable is a formula. * Given a formula ''X'', the

negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...

''¬X'' is a formula. * Given two formulas ''X'' and ''Y'', and a

binary 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 ...

''b'' (such as the logical conjunction ∧),the expression ''(X b Y)'' is a formula. (Note the parentheses.) Through this construction, all of the formulas of propositional logic can be built up from propositional variables as a basic unit. Propositional variables should not be confused with the metavariables, which appear in the typical axioms of propositional calculus; the latter effectively range over well-formed formulae, and are often denoted using lower-case greek letters such as

\alpha

\beta

and

\gamma

Predicate logic

Propositional variables with no object variables such as ''x'' and ''y'' attached to predicate letters such as P''x'' and ''x''R''y'', having instead individual constants ''a'', ''b'', ..attached to predicate letters are propositional constants P''a'', ''a''R''b''. These propositional constants are atomic propositions, not containing propositional operators. The internal structure of propositional variables contains predicate letters such as P and Q, in association with bound individual variables (e.g., x, ''y''), individual constants such as ''a'' and ''b'' ( singular terms from a domain of discourse D), ultimately taking a form such as P''a'', ''a''R''b''.(or with parenthesis,

P(11)

and

R(1, 3)

). Propositional logic is sometimes called

zeroth-order logic Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,. but an alternative definition extends propositional logic by adding constan ...

due to not considering the internal structure in contrast with

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

which analyzes the internal structure of the atomic sentences.

References

Bibliography

* Smullyan, Raymond M. ''First-Order Logic''. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic. {{logic-stub Propositional calculus Concepts in logic Logic symbols

Uses

Predicate logic

See also

References

Bibliography