natural deduction - définition. Qu'est-ce que natural deduction
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Qu'est-ce (qui) est natural deduction - définition

KIND OF PROOF CALCULUS
Introduction rule; Elimination rule; Natural deduction logic; Natural deduction calculus; Natural deduction system; Natural deductive logic

natural deduction         
<logic> A set of rules expressing how valid proofs may be constructed in predicate logic. In the traditional notation, a horizontal line separates premises (above) from conclusions (below). Vertical ellipsis (dots) stand for a series of applications of the rules. "T" is the constant "true" and "F" is the constant "false" (sometimes written with a LaTeX perp). "^" is the AND (conjunction) operator, "v" is the inclusive OR (disjunction) operator and "/" is NOT (negation or complement, normally written with a LaTeX eg). P, Q, P1, P2, etc. stand for propositions such as "Socrates was a man". P[x] is a proposition possibly containing instances of the variable x, e.g. "x can fly". A proof (a sequence of applications of the rules) may be enclosed in a box. A boxed proof produces conclusions that are only valid given the assumptions made inside the box, however, the proof demonstrates certain relationships which are valid outside the box. For example, the box below labelled "Implication introduction" starts by assuming P, which need not be a true proposition so long as it can be used to derive Q. Truth introduction: - T (Truth is free). Binary AND introduction: ----------- | . | . | | . | . | | Q1 | Q2 | ----------- Q1 ^ Q2 (If we can derive both Q1 and Q2 then Q1^Q2 is true). N-ary AND introduction: ---------------- | . | .. | . | | . | .. | . | | Q1 | .. | Qn | ---------------- Q1^..^Qi^..^Qn Other n-ary rules follow the binary versions similarly. Quantified AND introduction: --------- | x . | | . | | Q[x] | --------- For all x . Q[x] (If we can prove Q for arbitrary x then Q is true for all x). Falsity elimination: F - Q (Falsity opens the floodgates). OR elimination: P1 v P2 ----------- | P1 | P2 | | . | . | | . | . | | Q | Q | ----------- Q (Given P1 v P2, if Q follows from both then Q is true). Exists elimination: Exists x . P[x] ----------- | x P[x] | | . | | . | | Q | ----------- Q (If Q follows from P[x] for arbitrary x and such an x exists then Q is true). OR introduction 1: P1 ------- P1 v P2 (If P1 is true then P1 OR anything is true). OR introduction 2: P2 ------- P1 v P2 (If P2 is true then anything OR P2 is true). Similar symmetries apply to ^ rules. Exists introduction: P[a] ------------- Exists x.P[x] (If P is true for "a" then it is true for all x). AND elimination 1: P1 ^ P2 ------- P1 (If P1 and P2 are true then P1 is true). For all elimination: For all x . P[x] ---------------- P[a] (If P is true for all x then it is true for "a"). For all implication introduction: ----------- | x P[x] | | . | | . | | Q[x] | ----------- For all x . P[x] -> Q[x] (If Q follows from P for arbitrary x then Q follows from P for all x). Implication introduction: ----- | P | | . | | . | | Q | ----- P -> Q (If Q follows from P then P implies Q). NOT introduction: ----- | P | | . | | . | | F | ----- / P (If falsity follows from P then P is false). NOT-NOT: //P --- P (If it is not the case that P is not true then P is true). For all implies exists: P[a] For all x . P[x] -> Q[x] ------------------------------- Q[a] (If P is true for given "a" and P implies Q for all x then Q is true for a). Implication elimination, modus ponens: P P -> Q ---------- Q (If P and P implies Q then Q). NOT elimination, contradiction: P /P ------ F (If P is true and P is not true then false is true). (1995-01-16)
Natural deduction         
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Natural disaster         
  • 1755 copper engraving depicting [[Lisbon]] in ruins and in flames after the [[1755 Lisbon earthquake]]. A [[tsunami]] overwhelms the ships in the harbor.
  • A classic anvil-shaped, and clearly-developed [[Cumulonimbus incus]]
  • Global damage cost from natural disasters
  • Global Number of deaths from earthquake (1960-2017)
  • A [[blizzard]] in [[Maryland]] in 2009
  • frac=2}} in diameter
  • The [[Limpopo River]] during the [[2000 Mozambique flood]]
  • Global death from natural disasters
  • Global number of recorded earthquake events
  • A rope [[tornado]] in its dissipating stage, [[Tecumseh, Oklahoma]].
  • wildfire]] in [[California]].
  • A landslide in [[San Clemente, California]] in 1966
MAJOR ADVERSE EVENT RESULTING FROM NATURAL PROCESSES OF THE EARTH, WHICH MAY CAUSE LOSS OF LIFE OR PROPERTY
Meteorological disasters; Natural Disasters; Natural disasters; Natural Disaster; Weather disasters; Examples of natural disaster; Natural Disaster (song); Hydrological disaster; Hydrological disasters; Meteorological disaster; Political effects of natural disasters
A natural disaster is a major adverse event resulting from natural processes of the Earth; examples include firestorms, duststorms, floods, hurricanes, tornadoes, volcanic eruptions, earthquakes, tsunamis, storms, and other geologic processes. A natural disaster can cause loss of life or damage property, and typically leaves some economic damage in its wake, the severity of which depends on the affected population's resilience and on the infrastructure available.

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Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.