Logical expressions are expressions that evaluate to a boolean value, which can be either true or false. In programming, mathematics, and philosophy, logical expressions are used to make decisions, perform operations, and evaluate conditions. ### Components of Logical Expressions: 1. **Operands:** The variables or values being evaluated. For example, in the expression `A AND B`, `A` and `B` are operands. 2. **Operators:** The symbols that represent logical operations.
An atomic formula, in the context of formal logic and mathematical logic, is a basic type of formula that expresses a simple statement or proposition about a specific relation or property without any logical connectives (such as AND, OR, NOT, etc.). An atomic formula typically consists of: 1. **Predicate Symbols**: These are symbols that represent properties or relations.
Cirquent calculus is a formal system that extends the traditional sequent calculus, aiming to handle certain aspects of logic more effectively, particularly in the context of proof theory and structural proof theories. The main innovation in cirquent calculus is its ability to represent proofs in a more flexible way by using what are called "cirquents." A cirquent is a generalization of a sequent, allowing for multiple premises and conclusions that can be structured in a graph-like form rather than in a linear sequence.
The term "open formula" can refer to different concepts depending on the context in which it is used: 1. **Mathematics/Logic**: In mathematical logic, an open formula is a predicate that contains free variables. Unlike closed formulas (which are universal statements that can be evaluated as true or false), open formulas depend on the values assigned to their free variables.
Polish notation, also known as prefix notation, is a mathematical notation in which the operator precedes its operands. This means that instead of writing an expression in the conventional infix notation (where operators are placed between operands), Polish notation allows for expressions to be written without the need for parentheses to denote order of operations.
"Sequent" can refer to different concepts depending on the context. Here are a few possibilities: 1. **Sequent Calculus**: In mathematical logic, a sequent is a formal expression used in sequent calculus, which is a type of proof system.
In mathematics and logic, a theorem is a statement or proposition that has been proven to be true based on previously established statements, such as axioms, definitions, and previously proven theorems. The proof of a theorem typically involves deductive reasoning and follows a logical framework. The process of proving a theorem ensures that it holds under the conditions specified and contributes to the broader body of knowledge within a particular mathematical discipline or logical system.
In mathematical logic, a **theory** is a formal system that consists of a set of sentences or propositions in a particular language, along with a set of axioms and inference rules that determine what can be derived or proven within that system. The sentences are typically formulated in first-order logic or another formal logical language, and they can express various mathematical statements or properties.
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