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 artificial intelligence, the frame problem describes an issue with using first-order logic (FOL) to express facts about a robot in the world. Representing the state of a robot with traditional FOL requires the use of many axioms that simply imply that things in the environment do not change arbitrarily. For example, Hayes describes a "block world" with rules about stacking blocks together. In a FOL system, additional axioms are required to make inferences about the environment. The frame problem is the problem of finding adequate collections of axioms for a viable description of a robot environment.
Natural language processing (NLP) is a subfield of linguistics, computer science, information engineering, and artificial intelligence concerned with the interactions between computers and human (natural) languages, in particular how to program computers to process and analyze large amounts of natural language data.
Prolog is a logic programming language associated with artificial intelligence and computational linguistics.
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.
Truth is the property of being in accord with fact or reality. In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences.
In philosophy, vagueness is an important problem in semantics, metaphysics and philosophical logic. Definitions of this problem vary. A predicate is vague if it has borderline cases. The predicate "is tall" is vague because there seems to be no particular height at which someone becomes tall. Alternately, a predicate is sometimes said to be vague if there are borderline cases of its application, such that in these cases competent speakers of the language may faultlessly disagree over whether the predicate applies. The disagreement over whether a hotdog is a sandwich suggests that “sandwich” is vague.
The raven paradox, also known as Hempel's paradox, Hempel's ravens, or rarely the paradox of indoor ornithology, is a paradox arising from the question of what constitutes evidence for a statement. Observing objects that are neither black nor ravens may formally increase the likelihood that all ravens are black even though, intuitively, these observations are unrelated.
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, highlighting the apparent lack of justification for:
- Generalizing about the properties of a class of objects based on some number of observations of particular instances of that class or
- Presupposing that a sequence of events in the future will occur as it always has in the past. Hume called this the principle of uniformity of nature.
The Critique of Pure Reason is a book by the German philosopher Immanuel Kant, in which the author seeks to determine the limits and scope of metaphysics. Also referred to as Kant's "First Critique", it was followed by the Critique of Practical Reason (1788) and the Critique of Judgment (1790). In the preface to the first edition, Kant explains that by a "critique of pure reason" he means a critique "of the faculty of reason in general, in respect of all knowledge after which it may strive independently of all experience" and that he aims to reach a decision about "the possibility or impossibility of metaphysics."
Nicholas Rescher is a German-American philosopher, polymath, and author, teaching at the University of Pittsburgh. He is the Chairman of the Center for Philosophy of Science and has formerly served as Chairman of the Philosophy Department.
Grue and bleen are examples of logical predicates coined by Nelson Goodman in Fact, Fiction, and Forecast to illustrate the "new riddle of induction" – a successor to Hume's original problem. These predicates are unusual because their application is time-dependent; many have tried to solve the new riddle on those terms, but Hilary Putnam and others have argued such time-dependency depends on the language adopted, and in some languages it is equally true for natural-sounding predicates such as "green." For Goodman they illustrate the problem of projectible predicates and ultimately, which empirical generalizations are law-like and which are not. Goodman's construction and use of grue and bleen illustrates how philosophers use simple examples in conceptual analysis.
The Self-Indication Assumption Doomsday argument rebuttal is an objection to the Doomsday argument by arguing that the chance of being born is not one, but is an increasing function of the number of people who will be born.
Timothy Williamson is a British philosopher whose main research interests are in philosophical logic, philosophy of language, epistemology and metaphysics. He is the Wykeham Professor of Logic at the University of Oxford, and fellow of New College, Oxford.
In computability theory, the T predicate, first studied by mathematician Stephen Cole Kleene, is a particular set of triples of natural numbers that is used to represent computable functions within formal theories of arithmetic. Informally, the T predicate tells whether a particular computer program will halt when run with a particular input, and the corresponding U function is used to obtain the results of the computation if the program does halt. As with the smn theorem, the original notation used by Kleene has become standard terminology for the concept.
Vagrancy is the condition of homelessness without regular employment or income. Vagrants usually live in poverty and support themselves by begging, garbage scraping, petty theft, temporary work, or welfare.
In natural languages, a quantifier turns a sentence about something having some property into a sentence about the number (quantity) of things having the property. Examples of quantifiers in English are "all", "some", "many", "few", "most", and "no"; examples of quantified sentences are "all people are mortal", "some people are mortal", and "no people are mortal", they are considered to be true, true, and false, respectively.
Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.
Revision theory is a subfield of philosophical logic. It consists of a general theory of definitions, including circular and interdependent concepts. A circular definition is one in which the concept being defined occurs in the statement defining it—for example, defining a G as being blue and to the left of a G. Revision theory provides formal semantics for defined expressions, and formal proof systems study the logic of circular expressions.
