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The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
This system permits a test that compares how the gravitational pull of the outer white dwarf affects the pulsar, which has strong self-gravity, and the inner white dwarf. The result shows that the accelerations of the pulsar and its nearby white-dwarf companion differ fractionally by no more than 2.6 × 10 −6 (95% confidence level). [123 ...
10 −1 g dg decigram 10 1 g dag decagram 10 −2 g cg: centigram: 10 2 g hg hectogram 10 −3 g mg: milligram: 10 3 g kg: kilogram: 10 −6 g μg: microgram (mcg) 10 6 g Mg megagram 10 −9 g ng: nanogram: 10 9 g Gg gigagram 10 −12 g pg picogram 10 12 g Tg teragram 10 −15 g fg femtogram 10 15 g Pg petagram 10 −18 g ag attogram 10 18 g Eg ...
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is ...
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...
Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .
Then the attraction force vector onto a sample mass can be expressed as: F = m g {\displaystyle \mathbf {F} =m\mathbf {g} } Here g {\displaystyle \mathbf {g} } is the frictionless , free-fall acceleration sustained by the sampling mass m {\displaystyle m} under the attraction of the gravitational source.
It is the inference that, if P implies Q and R implies S and either Q is false or S is false, then either P or R must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. Destructive dilemma is the disjunctive version of modus tollens.