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 Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, or alternative denial since it says in effect that at least one of its operands is false, or NAND. In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as or as or as or as in Polish notation by Łukasiewicz.
A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range of topics in artificial intelligence, computer science, and cognitive science.
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 abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature
In proof theory, the semantic tableau (; plural: tableaux, also called an analytic tableau, truth tree, or simply tree, is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula. The tableau method can also determine the satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure for modal logics.
In mathematical logic, a propositional variable is an input variable of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.
A logical graph is a special type of diagrammatic structure of graphical syntax developed for logic.
An entitative graph is an element of the diagrammatic syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic beginning in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. See 3.468, 4.434, and 4.564 in Peirce's Collected Papers.
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.
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 mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.
Charles Sanders Peirce began writing on semiotics, which he also called semeiotics, meaning the philosophical study of signs, in the 1860s, around the time that he devised his system of three categories. During the 20th century, the term "semiotics" was adopted to cover all tendencies of sign researches, including Ferdinand de Saussure's semiology, which began in linguistics as a completely separate tradition.
In database theory, a conjunctive query is a restricted form of first-order queries using the logical conjunction operator. Many first-order queries can be written as conjunctive queries. In particular, a large part of queries issued on relational databases can be expressed in this way. Conjunctive queries also have a number of desirable theoretical properties that larger classes of queries do not share.
On May 14, 1867, the 27–year-old Charles Sanders Peirce, who eventually founded pragmatism, presented a paper entitled "On a New List of Categories" to the American Academy of Arts and Sciences. Among other things, this paper outlined a theory of predication involving three universal categories that Peirce continued to apply in philosophy and elsewhere for the rest of his life. The categories demonstrate and concentrate the pattern seen in "How to Make Our Ideas Clear", and other three-way distinctions in Peirce's work.
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors that operate on terms to yield terms. PFL is mostly the invention of the logician and philosopher Willard Quine.
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
