In mathematical logic, a propositional variable (also called a sentential variable) is a variable which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics. Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ...
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. For example, in a given propositional logic, we might define a formula as follows: In formal logic, logical connectives, also known as logical connectors and sometimes logical constants, serve to connect statements into more complicated compound statements. ... In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ...
Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), then XbY is a formula.
In this way, all of the formulas of propositional logic are built up from propositional variables as a basic unit. Negation (i. ... AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ...
References
Smullyan, Raymond M. First-Order Logic. 1968. Dover edition, 1995. Chapter 1.1: Formulas of Propositional Logic.
The logical connectives or logical constants for "and," for "or," for "implies," and for "not"" are used to connect propositionalvariables to form compounds of propositional forms.
A single proposition can be true or false, two truth-values, but a compound proposition of two simple propositions has four possibilities: both can be true; the first true and the second false; the first false and the second true; and both can be false.
The propositionalvariables are "c," "b," and "w." c = cat is meowing; b = baby is crying; w = wind is blowing.
Propositional logic is a domain of formal subject matter that is, up to isomorphism, constituted by the structural relationships of mathematical objects called propositions.
The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and (2) a set of operator symbols, variously interpreted as logical operators or logical connectives.
When the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, they yield first-order logic, or first-order predicate logic, which keeps all the rules of propositional logic and adds some new ones.
is a formula.
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