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In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.
Saying that a sentence is algebraic is a stronger condition than saying it is elementary.
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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.
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 quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.
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 and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In mathematics, model theory is the study of classes of mathematical structures from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a theory. A model of a theory is a structure that satisfies the sentences of that theory.
In logic, the semantic principleof bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.
Classical logic is the intensively studied and most widely used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.
Laws of Form is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. LoF describes three distinct logical systems:
In mathematical logic, the Lindenbaum–Tarski algebra of a logical theory T consists of the equivalence classes of sentences of the theory. That is, two sentences are equivalent if the theory T proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation.
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.
A Boolean-valued function is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set,, whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.
In mathematics and philosophy, Łukasiewicz logic is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic; it was later generalized to n-valued as well as infinitely-many-valued (ℵ0-valued) variants, both propositional and first-order. The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz–Tarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics.
In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics.
In mathematical logic, an algebraic definition is one that can be given using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed.
Informally in mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.
In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory.
In mathematical logic, an elementary theory is one that involves axioms using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory.
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction (and) denoted as ∧, the disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. It is thus a formalism for describing logical operations in the same way that elementary algebra describes numerical operations.