In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
In logic and related fields such as mathematics and philosophy, "if and only if" is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional, and can be likened to the standard material conditional combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other, though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false.
A false dilemma, also referred to as false dichotomy or false binary, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false premise. This premise has the form of a disjunctive claim: it asserts that one among a number of alternatives must be true. This disjunction is problematic because it oversimplifies the choice by excluding viable alternatives, presenting the viewer with only two absolute choices when in fact, there could be many.
Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.
Causality is an influence by which one event, process, state, or object (acause) contributes to the production of another event, process, state, or object (an effect) where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The idea that "correlation implies causation" is an example of a questionable-cause logical fallacy, in which two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known by the Latin phrase cum hoc ergo propter hoc. This differs from the fallacy known as post hoc ergo propter hoc, in which an event following another is seen as a necessary consequence of the former event, and from conflation, the errant merging of two events, ideas, databases, etc., into one.
In classical rhetoric and logic, begging the question or assuming the conclusion is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion. Historically, begging the question refers to a fault in a dialectical argument in which the speaker assumes some premise that has not been demonstrated to be true. In modern usage, it has come to refer to an argument in which the premises assume the conclusion without supporting it. This makes it more or less synonymous with circular reasoning.
A fallacy, is the use of invalid or otherwise faulty reasoning in the construction of an argument that may appear to be well-reasoned if unnoticed. The term was introduced in the Western intellectual tradition by the Aristotelian De Sophisticis Elenchis.
In computer programming, the exclusive or swap is an algorithm that uses the exclusive or bitwise operation to swap the values of two variables without using the temporary variable which is normally required.
Post hoc ergo propter hoc is an informal fallacy that states: "Since event Y followed event X, event Y must have been caused by event X." It is often shortened simply to post hoc fallacy. A logical fallacy of the questionable cause variety, it is subtly different from the fallacy cum hoc ergo propter hoc, in which two events occur simultaneously or the chronological ordering is insignificant or unknown. Post hoc is a logical fallacy in which one event seems to be the cause of a later event because it occurred earlier.
A faulty generalization is an informal fallacy wherein a conclusion is drawn about all or many instances of a phenomenon on the basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an example of jumping to conclusions. For example, one may generalize about all people or all members of a group from what one knows about just one or a few people:
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form (CDNF) or minterm canonical form, and its dual, the canonical conjunctive normal form (CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form, and the algebraic normal form.
Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the form of the argument, as is the case for formal fallacies, but can also be due to their content and context. Fallacies, despite being incorrect, usually appear to be correct and thereby can seduce people into accepting and using them. These misleading appearances are often connected to various aspects of natural language, such as ambiguous or vague expressions, or the assumption of implicit premises instead of making them explicit.
In philosophical logic, the masked-man fallacy is committed when one makes an illicit use of Leibniz's law in an argument. Leibniz's law states that if A and B are the same object, then A and B are indiscernible. By modus tollens, this means that if one object has a certain property, while another object does not have the same property, the two objects cannot be identical. The fallacy is "epistemic" because it posits an immediate identity between a subject's knowledge of an object with the object itself, failing to recognize that Leibniz's Law is not capable of accounting for intensional contexts.
In causal inference, a confounder is a variable that influences both the dependent variable and independent variable, causing a spurious association. Confounding is a causal concept, and as such, cannot be described in terms of correlations or associations. The existence of confounders is an important quantitative explanation why correlation does not imply causation. Some notations are explicitly designed to identify the existence, possible existence, or non-existence of confounders in causal relationships between elements of a system.
A fallacy of necessity is a fallacy in the logic of a syllogism whereby a degree of unwarranted necessity is placed in the conclusion.
In logic and philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
Physical causal closure is a metaphysical theory about the nature of causation in the physical realm with significant ramifications in the study of metaphysics and the mind. In a strongly stated version, physical causal closure says that "all physical states have pure physical causes" — Jaegwon Kim, or that "physical effects have only physical causes" — Agustin Vincente, p. 150.
In the philosophy of science, a causal model is a conceptual model that describes the causal mechanisms of a system. Several types of causal notation may be used in the development of a causal model. Causal models can improve study designs by providing clear rules for deciding which independent variables need to be included/controlled for.
