Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Proof assistantIn computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.
ContrapositionIn logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Conditional statement . In formulas: the contrapositive of is . If P, Then Q. — If not Q, Then not P. "If it is raining, then I wear my coat" — "If I don't wear my coat, then it isn't raining.
Logical truthLogical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants). Thus, logical truths such as "if p, then p" can be considered tautologies.
Mathematical logicMathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
AngleIn Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.
CausalityCausality (also called causation, or cause and effect) is influence by which one event, process, state, or object (a cause) 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.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
Term logicIn logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century.
Propositional attitudeA propositional attitude is a mental state held by an agent or organism toward a proposition. In philosophy, propositional attitudes can be considered to be, neurally-realized causally efficacious content-bearing internal states (personal principles/values). Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won'. Propositional attitudes are often assumed to be the fundamental units of thought and their contents, being propositions, are true or false from the perspective of the person.
Naive set theoryNaive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.
Philosophy of logicPhilosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines.
Intuitionistic logicIntuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L.
Evidence-based policyEvidence-based policy is a concept in public policy that advocates for policy decisions to be grounded on, or influenced by, rigorously established objective evidence. This concept presents a stark contrast to policymaking predicated on ideology, 'common sense,' anecdotes, or personal intuitions. The approach mirrors the effective altruism movement's philosophy within governmental circles. The methodology employed in evidence-based policy often includes comprehensive research methods such as randomized controlled trials (RCT).
Angle trisectionAngle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle (that is, to construct an angle of 30 degrees).
Internal and external anglesIn geometry, an angle of a polygon is formed by two adjacent sides. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex.
Evidence-based practiceEvidence-based practice (EBP) is the idea that occupational practices ought to be based on scientific evidence. While seemingly obviously desirable, the proposal has been controversial, with some arguing that results may not specialize to individuals as well as traditional practices. Evidence-based practices have been gaining ground since the formal introduction of evidence-based medicine in 1992 and have spread to the allied health professions, education, management, law, public policy, architecture, and other fields.
Transposition (logic)In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that where "" is a metalogical symbol representing "can be replaced in a proof with".
Garden toolA garden tool is any one of many tools made for gardening and landscaping, which overlap with the range of tools made for agriculture and horticulture. Garden tools can be divided into hand tools and power tools. Hand tool Today's garden tools originated with the earliest agricultural implements used by humans. Examples include the hatchet, axe, sickle, scythe, pitchfork, spade, shovel, trowel, hoe, fork, and rake. In some places, the machete is common. The earliest tools were made variously of wood, flint, metal, tin, and bone.
Hand toolA hand tool is any tool that is powered by hand rather than a motor. Categories of hand tools include wrenches, pliers, cutters, , striking tools, struck or hammered tools, screwdrivers, vises, clamps, snips, hacksaws, drills, and knives. Outdoor tools such as garden forks, pruning shears, and rakes are additional forms of hand tools. Portable power tools are not hand tools. Hand tools have been used by humans since the Stone Age when stone tools were used for hammering and cutting.
