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The up tack or falsum (⊥, \bot in LaTeX, U+22A5 in Unicode [1]) is a constant symbol used to represent: . The truth value 'false', or a logical constant denoting a proposition in logic that is always false (often called "falsum" or "absurdum").
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
Isaac Barrow and Baermann used the notation Q.E.A., for "quod est absurdum" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today. [12] A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley. [13]
Zeno's arguments may then be early examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one."
The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area.
Uncle Joe concludes his proof reductio ad absurdum, meaning in English "proof by contradiction". What Carroll calls the protasis of a conditional is now known as the antecedent, and similarly the apodosis is now called the consequent. Symbols can be used to greatly simplify logical statements such as those inherent in this story:
Gödel noted that each statement within a system can be represented by a natural number (its Gödel number).The significance of this was that properties of a statement—such as its truth or falsehood—would be equivalent to determining whether its Gödel number had certain properties.
The heat death paradox, also known as thermodynamic paradox, Clausius' paradox, and Kelvin's paradox, [1] is a reductio ad absurdum argument that uses thermodynamics to show the impossibility of an infinitely old universe.