Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist Abhidharma schools (mainly Vaibhasika) which posited theories of svabhava (essential nature) and also the Hindu Nyāya and Vaiśeṣika schools which posited a theory of ontological substances ( dravyatas). In the Mūlamadhyamakakārikā, Nāgārjuna's reductio ad absurdum arguments are used to show that any theory of substance or essence was unsustainable and therefore, phenomena ( dharmas) such as change, causality, and sense perception were empty ( sunya) of any essential existence. Much of Madhyamaka Buddhist philosophy centers on showing how various essentialist ideas have absurd conclusions through reductio ad absurdum arguments (known as prasaṅga, "consequence" in Sanskrit). Īnother example of this technique is found in the sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap. The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as ( Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b). In response, Socrates, via a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion and adopt a position of aporia. Typically, Socrates' opponent would make what would seem to be an innocuous assertion. The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method ( elenchus), also called the Socratic method. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples. Greek mathematicians proved fundamental propositions using reductio ad absurdum. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false. The gods cannot have both forms, so this is a contradiction. But if horses and oxen could draw, they would draw the gods with horse and ox bodies. Criticizing Homer's attribution of human faults to the gods, Xenophanes states that humans also believe that the gods' bodies have human form. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. Reductio ad absurdum was used throughout Greek philosophy. The second example is a mathematical proof by contradiction (also known as an indirect proof ), which argues that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it). The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one.The Earth cannot be flat otherwise, since the Earth is assumed to be finite in extent, we would find people falling off the edge.The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show: This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. 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. Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884.
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