Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as , and .[1]
Example
In a given propositional logic, a formula can be defined as follows:
Given two formulas X and Y, and a binary connectiveb (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 , and .
Predicate logicedit
Propositional variables with no object variables such as x and y attached to predicate letters such as Px and xRy, having instead individual constants a, b, ..attached to predicate letters are propositional constants Pa, aRb. 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 Pa, aRb.(or with parenthesis, and ).[2]
Propositional logic is sometimes called zeroth-order logic due to not considering the internal structure in contrast with first-order logic which analyzes the internal structure of the atomic sentences.
propositional, variable, mathematical, logic, propositional, variable, also, called, sentential, variable, sentential, letter, input, variable, that, either, true, false, truth, function, basic, building, blocks, propositional, formulas, used, propositional, l. In mathematical logic 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 and higher order logics Contents 1 Uses 2 Predicate logic 3 See also 4 References 5 BibliographyUses editFormulas in logic are typically built up recursively from some propositional variables some number of logical connectives and some logical quantifiers Propositional variables are the atomic formulas of propositional logic and are often denoted using capital roman letters such as P displaystyle P nbsp Q displaystyle Q nbsp and R displaystyle R nbsp 1 ExampleIn a given propositional logic a formula can be defined as follows Every propositional variable is a formula Given a formula X the negation X is a formula Given two formulas X and Y and a binary connective 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 a displaystyle alpha nbsp b displaystyle beta nbsp and g displaystyle gamma nbsp Predicate logic editPropositional variables with no object variables such as x and y attached to predicate letters such as Px and xRy having instead individual constants a b attached to predicate letters are propositional constants Pa aRb 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 Pa aRb or with parenthesis P 11 displaystyle P 11 nbsp and R 1 3 displaystyle R 1 3 nbsp 2 Propositional logic is sometimes called zeroth order logic due to not considering the internal structure in contrast with first order logic which analyzes the internal structure of the atomic sentences See also editBoolean algebra logic Boolean datatype Boolean domain Boolean functionLogical value Predicate variable Propositional logicReferences edit Predicate Logic Brilliant Math amp Science Wiki brilliant org Retrieved 2020 08 20 Mathematics Predicates and Quantifiers Set 1 GeeksforGeeks 2015 06 24 Retrieved 2020 08 20 Bibliography editSmullyan Raymond M First Order Logic 1968 Dover edition 1995 Chapter 1 1 Formulas of Propositional Logic nbsp This logic related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Propositional variable amp oldid 1062676408, wikipedia, wiki, book, books, library,
